High-Frequency Statistics with Asynchronous and Irregular Data [1st ed. 2019] 978-3-658-28417-6, 978-3-658-28418-3

Table of contents :
Front Matter ....Pages I-XIII
Front Matter ....Pages 1-1
Framework (Ole Martin)....Pages 3-8
Laws of Large Numbers (Ole Martin)....Pages 9-71
Central Limit Theorems (Ole Martin)....Pages 73-109
Estimating Asymptotic Laws (Ole Martin)....Pages 111-135
Observation Schemes (Ole Martin)....Pages 137-153
Front Matter ....Pages 155-155
Estimating Spot Volatility (Ole Martin)....Pages 157-174
Estimating Quadratic Covariation (Ole Martin)....Pages 175-205
Testing for the Presence of Jumps (Ole Martin)....Pages 207-240
Testing for the Presence of Common Jumps (Ole Martin)....Pages 241-303
Back Matter ....Pages 305-323

Citation preview

Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics

Ole Martin

High-Frequency Statistics with Asynchronous and Irregular Data

Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics Reihe herausgegeben von Ralf Werner, Augsburg, Deutschland Tobias Harks, Augsburg, Deutschland Vladimir Shikhman, Chemnitz, Deutschland

In der Reihe werden Arbeiten zu aktuellen Themen der mathematischen Optimie­ rung und der Wirtschaftsmathematik publiziert. Hierbei werden sowohl Themen aus Grundlagen, Theorie und Anwendung der Wirtschafts-, Finanz- und Ver­ sicherungsmathematik als auch der Optimierung und des Operations Research behandelt. Die Publikationen sollen insbesondere neue Impulse für weiterge­ hende Forschungsfragen liefern, oder auch zu offenen Problemstellungen aus der Anwendung Lösungsansätze anbieten. Die Reihe leistet damit einen Beitrag der Bündelung der Forschung der Optimierung und der Wirtschaftsmathematik und der sich ergebenden Perspektiven.

Weitere Bände in der Reihe http://www.springer.com/series/15822

Ole Martin

High-Frequency Statistics with Asynchronous and Irregular Data

Ole Martin Department of Mathematics Kiel University (CAU) Kiel, Germany Dissertation, Kiel University (CAU), 2018

ISSN 2523-7926 ISSN 2523-7934  (electronic) Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics ISBN 978-3-658-28418-3  (eBook) ISBN 978-3-658-28417-6 https://doi.org/10.1007/978-3-658-28418-3 Springer Spektrum © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer Spektrum imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Preface Over the past two decades the field of high-frequency statistics especially with applications to financial data experienced a rapid growth1 . This growth was fueled by the increasing availability of high-frequency data generated by electronic trading platforms and the rising importance of high-frequency traders. Various mathematical methods have therefore been designed to help practitioners as well as academics to investigate such data. The setting of high-frequency statistics is characterized as follows: We assume that we observe (multiple) stochastic pro(l) cess(es) (Xt )t≥0 , l = 1, . . . , d, which exist in continuous time over discrete grids (l)

of time points ti,n , i ∈ N. The goal then is to infer on properties of the continuoustime model based on the discrete observations X

(l) (l)

ti,n

, i ∈ N.

To this end we consider a fixed time interval [0, T ] and investigate the asymptotics (l) where the mesh of the observation times ti,n , i ∈ N, (i.e. the maximal distance of consecutive observation times) tends to zero. Although the number of observations cannot be controlled by the statistician and is in practice always finite, we assume that their density in [0, T ] is high enough such that asymptotic results approximate the finite case reasonably well. Indeed in financial applications, financial processes are often observed over intervals as short as seconds or even milliseconds. Quite naturally people first started to work with simple and regular models for the data. But as the field grew and practical applications presented further challenges the models also evolved. So people moved from models for continuous processes to models incorporating jumps, started to model market frictions by including micro-structure noise and also began to consider more complex models for the observation times. While there exists a vast amount of literature on (l) the situation of equidistant and synchronous observation times ti,n = i/n, see the monographs [30] and [3] for an overview, the situation of irregular and asynchronous observation times has been investigated less often because of its higher complexity. Due to the lack of methods based on irregularly observed data often methods which were designed for equidistant and synchronous data are also applied to real data which usually comes in irregular and asynchronous form. Here, these methods can’t be applied to the data directly, but the data has to be artificially synchronized beforehand. This means that not all available observations 1 compare

Page XVII in [3]

VI

Preface

are used but instead only the observations closest to the fixed marks iΔn , i ∈ N, enter the estimation. Here the new grid width Δn ≥ 0 is usually much larger than the length of the average observation interval in the original dataset. The newly obtained dataset is then almost synchronous and the observation times are almost equidistant with mesh Δn . This technique allows to use the methods developed for equidistant and synchronous observations also in the irregular and asynchronous setting as the error originating from the fact, that the modified sampling scheme is only almost equidistant and synchronous, is usually dominated by other approximation errors. The big disadvantage of this method, however, is that we lose a lot of data as we only use very few selected observations. Although there exist methods which synchronize the data more efficiently, compare the concept of ”refresh times” times used e.g. in [5] and [1], a synchronization step always leads to a reduction of the number of used data points. Thereby the effective sample size is diminished and hence estimators have a higher variance, asymptotic distributions are less accurate proxies for finite sample distributions and the power of tests is usually smaller than in the setting where we observe the processes equally often in an equidistant and synchronous manner. Further, it has been shown both empirically in [14] and demonstrated mathematically in [22] that this artificial synchronization of the observations of multiple processes may lead to estimation bias. For these reasons, it seems superior to work with methods based directly on all irregular and asynchronous observations. Such methods for the estimation of the quadratic covariation were first presented in [22] and have been further investigated in [23], [24] and [20]. These methods have been extended to the discontinuous case where the processes are allowed to have jumps in the paper [8] for the estimation of the quadratic covariation including jumps and in [35], [34] for the estimation of the continuous part of the quadratic covariation under the presence of jumps. In this work, we generalize the methods developed in these papers for the estimation of the quadratic covariation to obtain asymptotic results for more general power-variation type functionals. These results are then used to construct statistical tests which allow to decide whether (common) jumps are present in irregularly and asynchronously observed processes. Further, we present a new bootstrap technique which allows to estimate asymptotic variances in the central limit theorems for statistics based on irregular and asynchronous observations. We find that in general the behaviour of functionals based on irregular and asynchronous observations is not only more complicated but sometimes even fundamentally different compared to the simpler setting of equidistant and synchronous observation times; see especially the results in Chapter 2. Further we will see that it becomes in general infeasible to use certain central limit theorems in the setting of irregular observations as the asymptotic variance can in general not be estimated from the data anymore; see Chapters 3 and 4. To circumvent this problem additional assumptions have to be made on the observation scheme such

Preface

VII

that a bootstrap method can be applied. The simulation results in Chapters 8 and 9 show that the finite sample performance of our methods based on irregular and asynchronous observations is comparable to the finite sample performance of similar methods based on equidistant and synchronous observations. This is a remarkable result as in practice data usually comes in irregular and asynchronous form and therefore for the application of methods designed for equidistant and synchronous observations the data has to be artificially made equidistant and synchronous as described before. Hence, our methods perform as well as methods based on equidistant and synchronous observations for similar sample sizes but the effective sample size is usually much larger when using methods designed for irregular and asynchronous data. We conclude that although the methods become more complicated it is worthwhile to work with procedures based directly on irregular and asynchronous observations as these methods yield much better results in practical applications. This work is divided into two parts: Part I contains general theoretical results which in similar form already have proven to be useful in the context of highfrequency statistics and which will be used in the applications in Part II. In Chapter 1 we characterize the framework and especially the general model for the stochastic processes and the observation times in which we are going to derive results. In Chapter 2 generalizations of functionals to the setting of irregular and asynchronous observations are investigated which regularly occur in the context of high-frequency statistics. In Chapters 3 and 4 we discuss how to obtain central limit theorems for the functionals introduced in Chapter 2 and how to estimate the asymptotic variances in these central limit theorems. Although Chapters 3 and 4 contain only results for specific functionals we use these as an example to introduce general methods which then later are also used in the applications part for more elaborate statistics. Throughout Chapters 2–4 we try to highlight differences in the results for irregular and asynchronous observation times compared to existing results in the simpler setting of equidistant and synchronous observations. In Chapter 5 we collect some results which are necessary to prove that certain specific observation schemes, most importantly the one where observation times are generated by independent Poisson processes, fulfil the conditions which were made in Chapters 2–4. Despite the use of these results already in the previous chapters, I decided to gather them in a separate chapter as the results are rather technical and partially also of individual interest. Part II contains applications of the methods developed in Part I for the estimation of certain quantities and for the testing of specific hypotheses which are of direct interest to practitioners. Chapter 6 is devoted to the estimation of spot volatilities and correlations. Albeit these estimators are already used in Chapter 4 I decided to discuss them in Part II separately. Their investigation to me seems to fit better into Part II because their estimation has been discussed in the literature in several contexts individually

VIII

Preface

which makes the topic of their estimation worthy of an own chapter. In Chapter 7 we discuss the estimation of realized quadratic variation and covariation processes which is probably the quantity which has received the most attention in the highfrequency statistics literature. Chapters 8 and 9 contain tests for the presence of (common) jumps in paths of observed processes. This work is mainly based on the research published in the three papers [36], [38] and [37]. It aims at presenting the methods used and results found therein in a unifying context. Thereby I try to draw a comprehensive picture of our approach in addressing the challenges arising from the use of irregular and asynchronous data in the field of high-frequency statistics. In particular Sections 2.1 and 2.2 are almost entirely adopted from the paper [37]. Further, Section 9.1 is based on [36] and Chapter 8 and Section 9.2 contain the results presented in [38]. Additionally, parts of the results and proofs which make up the content of the remaining chapters and sections are also modified versions of results and proofs which are included in the papers mentioned above. I am very thankful to my doctoral advisor Prof. Dr. Mathias Vetter who gave me the opportunity to write this thesis and also for giving me a lot of freedom in choosing the direction of my research. Further, I would like to thank my colleagues from the working groups stochastics and financial mathematics at the CAU Kiel for providing a welcoming working environment and the participants of the Oberseminar Stochastik und Finanzmathematik“ for listening to presenta” tions on earlier stages of this work and for giving helpful comments. I am thankful to the organizers of the DynStoch meeting 2017 held in Siegen, to the organizers of the DynStoch meeting 2018 in Porto and to the organizing committee of the 13th German Probability and Statistics Days 2018 in Freiburg i. Br. for giving me the opportunity to present my work in front of experienced researchers. Additionally, I would like to thank two anonymous referees whose comments improved the paper [36] and also further research. Further, I am grateful to my parents Ulla and Ulrich who from an early childhood age on encouraged my interest in the field of mathematics and who have always supported me. Finally, I am deeply indebted to my girlfriend Kathrin who is always there for me. Kiel, Germany

Ole Martin

Contents I

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

2 Laws of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . 2.1 Non-Normalized functionals . . . . . . . . . . . . . . . . . . . 2.1.1 The Results . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 The Proofs . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Normalized Functionals . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Results . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Proofs . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Functionals of Truncated Increments . . . . . . . . . . . . . . 2.3.1 The Results . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The Proofs . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Functionals of Increments over Multiple Observation Intervals 2.4.1 The Results . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 The Proofs . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

3 Central Limit Theorems . . . . . . . . . . . . . . . . . . . . 3.1 Central Limit Theorem for Non-Normalized Functionals 3.1.1 The Results . . . . . . . . . . . . . . . . . . . . . 3.1.2 The Proofs . . . . . . . . . . . . . . . . . . . . . 3.2 Central Limit Theorem for Normalized Functionals . . .

. . . . .

. . . . .

. 73 . 74 . 79 . 85 . 102

. . . . .

. . . . .

. . . . .

9 10 11 18 27 30 37 55 56 61 67 69 71

4 Estimating Asymptotic Laws . . . . . . . . . . . . . . . . . . . . . . . 111 4.1 The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.2 The Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5 Observation Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.1 The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.2 The Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

X

Contents

II Applications . . . . . . . . . . . . . . . . . . . . . . . . . 155 6 Estimating Spot Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.1 The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.2 The Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 7 Estimating Quadratic Covariation . . . . . . . . . . . . . 7.1 Consistency Results . . . . . . . . . . . . . . . . . . 7.2 Central Limit Theorems . . . . . . . . . . . . . . . . 7.3 Constructing Asymptotic Confidence Intervals . . . . 7.4 The Proofs . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

175 176 178 186 192

8 Testing for the Presence of Jumps 8.1 Theoretical Results . . . . . . 8.2 Simulation Results . . . . . . 8.3 The Proofs . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

207 208 220 228

9 Testing for the Presence of Common Jumps . . . . . . . 9.1 Null Hypothesis of Disjoint Jumps . . . . . . . . . . 9.1.1 Theoretical Results . . . . . . . . . . . . . . . 9.1.2 Simulation Results . . . . . . . . . . . . . . . 9.1.3 The Proofs . . . . . . . . . . . . . . . . . . . 9.2 Null Hypothesis of Joint Jumps . . . . . . . . . . . . 9.2.1 Theoretical Results . . . . . . . . . . . . . . . 9.2.2 Simulation Results . . . . . . . . . . . . . . . 9.2.3 The Proofs . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

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. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

241 242 242 253 257 267 267 279 287

. . . .

. . . .

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 A Estimates for Itˆ o Semimartingales . . . . . . . . . . . . . . . . . . . . 311 B Stable Convergence in Law . . . . . . . . . . . . . . . . . . . . . . . . 315 C Triangular Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

Notation Basic mathematical notation R, R≥0 , Rd N, N0 |x|, y |A| v∗ x, x x ∧ y, x ∨ y x+ A ∪ B, A ∩ B, A \ B Ac

1A P

−→ L

The real numbers, the non-negative real numbers and the space of d-dimensional vectors with real entries The natural numbers without 0, the natural numbers including 0 The absolute value of x ∈ Rd , the Euclidean norm of y ∈ Rd The Lebesgue-measure of a set A ⊂ R The transpose of a vector v ∈ Rd The largest (smallest) natural number less (greater) or equal than x ∈ R The minimum, maximum of x, y ∈ R The positive part x ∨ 0 of x ∈ R The union, intersection and difference of two sets A, B The complement of a set A The indicator function of a set A Convergence in probability

L

−→, =

Convergence in law, identity in law

−→ [X, Y ]t

U[a, b] Exp(λ) f (x) = o(g(x)) f (x) = O(g(x))

Stable convergence in law, see Appendix A The covariation process of the stochastic processes X and Y The left limit lims↑t Xs of X at time t if it exists The jump Xt − Xt− of a process X at time t The d-dimensional centered normal distribution with covariance matrix Σ ∈ Rd×d The uniform distribution over the interval [a, b] The exponential distribution with parameter λ > 0 It holds f (x)/g(x) → 0 as x → x0 It holds lim supx→x0 |f (x)/g(x)| < ∞

Xn = oP (Yn ) Xn = OP (Yn )

It holds Xn /Yn −→ 0 The sequence Xn /Yn is bounded in probability

L−s

Xt− ΔXt N (0, Σ)

P

XII

Notation

Repeatedly used variables in alphabetical order γ(z), Γt

Function γ and locally bounded proces Γt such that δ(t, z) ≤ Γt γ(z), see Condition 1.3 The function relating the jump measure μ to the jump sizes of X , see (1.1)

δ(s, z) (l)

(l)

Δi,n Y , Δi,k,n Y μ, ν πn |πn |t ρ ρ˜n (s, −), ρˆn (s, −), ρ˜n (s, +), ρˆn (s, +) σ, σ (l) (l)

(l)

The increments of a process Y over the inter(l) (l) vals Ii,n , Ii,k,n ; see (1.6) and (2.87) The Poisson random measure driving the jump part of X and its compensator, see (1.1) The set of all observation times at stage n, see (1.3) The mesh of the observation times, see (1.4) The correlation process of the continuous martingale parts of X (1) and X (2) , see (1.2) Estimators for the spot correlation of C (1) and C (2) at time s, see Chapter 6 The volatility matrix process of X, the volatility process of X (l) , see (1.2)

ˆn (s, −), σ ˜n (s, −), σ (l) (l) σ ˜n (s, +), σ ˆn (s, +)

Estimators for the spot volatility of X (l) at time s, see Chapter 6

τn,− (s), τn,+ (s)

The observation times of X (l) at stage n immeadiately before and after s, see (2.28) Probability space on which the process X and the observation times are defined Extended probability space on which limits of stable limit theorems can be defined The continuous part of finite variation in the decomposition of X, see (1.7) Sums of function evaluations at the jumps of X, X (l) ; see (2.5) The spot covariance matrix cs = σs σs∗ of C (1) , C (2) The continuous martingale part in the decomposition of X, see (1.7)

(l)

(l)

(Ω, F , P)

)  F, P (Ω, B(q) = (B (1) (q), B (2) (q))∗ B(f )T , B ∗ (f )T , B (l) (g)T cs C = (C (1) , C (2) )∗ (l),n

Gp (t), Gn p1 ,p2 (t), n Hk,m,p (t)

Sums of products of observation interval lengths occuring in the limits of the normalized functionals, see (2.39)

Notation

XIII

[k],(l),n

Gp

[k],n

(t), Gp1 ,p2 (t),

[k],n Hι,m,p (t) (l) in (s) (l)

Sums of products of cummulative lengths of multiple observation intervals, see (2.89) The index of the observation interval character(l) ized by s ∈ I (l) in (s),n

(l)

(l)

(l)

Observation interval (ti−1,n , ti,n ] of X (l) , inter-

Ii,n , Ii,k,n

(l)

K, Kq , Kp mΣ (h) M (q) = (M (1) (q), M (2) (q))∗ N (q) = (N (1) (q), N (2) (q))∗ S (l)

(l)

val (ti−k,n , ti,n ]; see (1.5) and (2.86) Generic constants (depending on q respectively p) Moment E[h(Z)] for Z ∼ N (0, Σ), see (2.38) The martingale part containing small jumps in the decomposition of X, see (1.7) The part containing big jumps in the decomposition of X, see (1.7) The σ-algebra generated by the observation times, see Definition 1.2

ti,n T V (f, πn )T , V (l) (g, πn )T

The i-th observation time of X (l) at stage n The time horizon T ≥ 0 Non-normalized functionals, see Section 2.1

V (p, f, πn )T , V (p, g, πn )T V+ (f, πn , (β, ))T , V − (p, f, πn , (β, ))T

Normalized functionals, see Section 2.2 Functionals of truncated increments, see (l) Section 2.3, also: V+ (g, πn , (β, ))T ,

V (f, [k], πn )T , V (p, f, [k], πn )T

V − (p, g, πn , (β, ))T Functionals of increments over multiple observation intervals, see Section 2.4, also:

(l)

(l)

X = (X X

(1)

,X

(l)

(2) ∗

)

V (l) (g, [k], πn )T , V (p, g, [k], πn )T The bivariate observed process The σ-algebra generated by the process X and its components, see Definition 1.2

Part I

Theory

1 Framework In this chapter, we specify the mathematical framework within we will derive theoretical results and develop statistical procedures. First, we characterize the stochastic processes which we observe. To this end (1) (2) let Xt = (Xt , Xt )∗ , t ≥ 0, be a stochastic process in continuous time. We restrict ourselves to bivariate processes as the case d = 2 is sufficient to study most effects of asynchronous observations and because a lot of questions on the dependence of multiple processes can be answered by investigating pairs of processes. Additionally, it is already in the bivariate case challenging to derive certain results. A common class of processes used to model dynamics in continuous time are semimartingales i.e. processes X which can be written as Xt = At + Mt where A is a process with paths of finite variation and M is a martingale. Semimartingales form the largest class of processes with respect to which a nice integration theory can be defined, compare page 35 in [3]. Thereby they naturally occur as solutions to stochastic differential equations. Further it has been shown that in financial mathematics price processes under certain assumptions have to be semimartingales; compare [13]. In this work we will not derive a theory for the whole class of semimartingales but only for the subclass of Itˆ o semimartingales. Itˆ o semimartingales can be understood as a generalization of L´evy processes where the L´evy-Khintchine triplet is allowed to be time dependent. Itˆ o semimartingales X distinguish themselves from general semimartingales by the property that it holds P(ΔXt = 0) = 0 for any t ≥ 0 where ΔXt = Xt − Xt− , Xt− = lims↑t Xs , denotes the jump of X at time t, compare Section 1.4.1 of [3]. This property implies that Itˆ o semimartingales have no fixed jump times. All Itˆ o semimartingales can be written in the so-called Grigelionis representation, compare Section 1.4.3 in [3], which we use in the following to give a more precise definition of the class of Itˆ o semimartingales. We denote by (Ω, (Ft )t≥0 , P) the filtered probability space on which the upcoming random variables will be defined. Definition 1.1. We call a process X = (X (1) , X (2) )∗ a two-dimensional Itˆ o semimartingale if it can be written in the form

t Xt = X0 +

t bs ds +

0

t  δ(s, z)1{δ(s,z)≤1} (μ − ν)(ds, dz)

σs dWs + 0

0 R2

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 O. Martin, High-Frequency Statistics with Asynchronous and Irregular Data, Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics, https://doi.org/10.1007/978-3-658-28418-3_1

4

1 Framework

t  δ(s, z)1{δ(s,z)>1} μ(ds, dz),

+

(1.1)

0 R2

where W is a two-dimensional standard Brownian motion, μ is a Poisson random measure on (0, ∞) × R2 whose predictable compensator satisfies the identity ν(ds, dz) = ds ⊗ λ(dz) for some σ-finite measure λ on R2 endowed with the Borelian σ-algebra. b is a two-dimensional adapted process, σ is a 2 × 2 adapted process and δ is a two-dimensional predictable function on Ω × (0, ∞) × R2 .  For a more detailed definition of the components of (1.1) and a detailed characterization of semimartingales and Itˆ o semimartingales in general we refer to Chapter 1 of [3]. As in Section 4 of [8] we further assume that σ is of the form

 σs =



(1)

σs

0



(2)

(1.2)

(2)

1 − ρ2s σs

ρs σs

(1)

(2)

for non-negative adapted processes σs , σs and an adapted process ρs with values in the interval [−1, 1]. This assumption is no additional restriction on the model because the law of (1.1) only depends on σt (σt )∗ and not on σt itself. If we are mainly interested in applications in finance, the decision to work only with Itˆ o semimartingales instead of general semimartingales is not very restrictive because according to [3] (page 1975, second paragraph) the assumption to work only with Itˆ o semimartingales is a mild structural assumption that is satisfied in ” all continuous-time models with stochastic volatility used in finance, at least as long as one wants to rule out arbitrage opportunities.“ Next we characterize the observation times: As data is usually aquired over time and previous states of a system normally can not be recovered later if the system has evolved over time it seems reasonable to assume that the observation times (l) ti,n are stopping times with respect to the filtration (Ft )t≥0 to which the process

X is adapted. At stage n we therefore assume that the process X (l) , l = 1, 2,  (l)  (l) is observed at the stopping times ti,n , i ∈ N0 . Further, ti,n i∈N , l = 1, 2, are 0

(l)

for each n ∈ N increasing sequences of stopping times and it holds t0,n = 0. We denote by πn =



(1) 

ti,n



i∈N0

(2) 

, ti,n



(1.3)

i∈N0

the collection of all observation times at stage n ∈ N which we will also call observation scheme and by



(l)

(l)



|πn |T = sup ti,n ∧ T − ti−1,n ∧ T i ≥ 1, l = 1, 2



(1.4)

1 Framework

5

≤ |πn |T

≤ |πn |T

≤ |πn |T

≤ |πn |T

X (1) (1)

(1) t0,n

(2) t0,n

=

(1)

t1,n

(1)

t2,n

(1)

t3,n

t4,n

=0

T (2)

(2)

t1,n

(2)

t2,n

(2)

t3,n

t4,n

X (2) ≤ |πn |T

≤ |πn |T

≤ |πn |T

≤ |πn |T

Figure 1.1: A realization of the observation scheme πn restricted to [0, T ]. the mesh of the observation times up to T . Based on the observation times we introduce some additional notation. By (l)

(l)

(l)

Ii,n = (ti−1,n , ti,n ]

(1.5)

we denote the i-th observation interval at stage n corresponding to the process X (l) . Further we define by (l)

Δi,n Y = Yt(l) − Yt(l) i,n

(1.6)

i−1,n

the increment of an arbitrary adapted process Y over the observation interval (l) (l) (l) Ii,n = (ti−1,n , ti,n ]. By |A| we denote the Lebesgue measure of a set A ⊂ [0, ∞) (l)

(l)

e.g. |Ii,n | is equal to the length of the observation interval Ii,n . The random (l)

(l)

variable in (s) is the index of the observation interval Ii,n which contains s i.e. (l)

in (s) is characterized via s ∈ I

(l)

(l)

in (s),n

. Throughout this book n is an unobservable

variable governing the observations and the asymptotics. If we allow the observation times to be arbitrary stopping times which may depend on the process X in an unspecified way it is very difficult to derive certain results. For this reason we will restrict ourselves very often to the setting of exogenous observation times i.e. observation times that are independent of the process X. Although this assumption seems to be rarely justified in practical applications it covers various interesting models and is far more general than the setting of equidistant and synchronous observation times usually considered in the literature. A more precise definition for exogeneity of the observation scheme is stated in the following.

6

1 Framework

Definition 1.2. Let S = σ({πn : n ∈ N}) denote the σ-algebra generated by the observation scheme and X = σ(X, b, σ, δ, W, μ) denote the σ-algebra generated by the process X and its components. We call an observation scheme (πn )n∈N exogenous if the observation scheme and the process X are independent, i.e. if S and X are independent.  To derive the upcoming results we have to impose some mild structural assumptions on the Itˆ o semimartingale (1.1) and the observation scheme, compare assumption (H) in [30] or [32], and we have to assume that the mesh |πn |T vanishes as n → ∞ because this is the property characterizing the asymptotics in the high-frequency setting. In particular if |πn |T does not vanish it is impossible to consistently infer on properties of the model like the jump behaviour which only become visible when considering the whole path in continuous time. These assumptions are summarized in the following condition. (1)

Condition 1.3. The process (bt )t≥0 is locally bounded and the processes (σt )t≥0 ,

(2) (σt )t≥0 , (ρt )t≥0 are c` adl` ag. Furthermore, Γt with δ(t, z) ≤ Γt γ(z) almost surely for 2

there exists a locally bounded process some deterministic bounded function (γ(z) ∧ 1)λ(dz) < ∞. The sequence of observation schemes

γ which satisfies (πn )n∈N fulfils

P

|πn |T −→ 0

(n → ∞).

In the decomposition (1.1) used in the definition of X small jumps and large jumps are distinguished by the property of whether or not their absolute size is less or equal respectively larger than 1. For q > 0 we additionally introduce the decomposition Xt = X0 + B(q)t + Ct + M (q)t + N (q)t of X, compare Appendix A in [8], where

 B(q)t =

 Ct =

t 0

 bs −



R2

(δ(s, z)1{δ(s,z)≤1} − δ(s, z)1{γ(z)≤1/q} )λ(dz) ds,

t 0

σs dWs ,

 t M (q)t =

0

R2

0

R2

 t N (q)t =

(1.7) δ(s, z)1{γ(z)≤1/q} (μ − ν)(ds, dz), δ(s, z)1{γ(z)>1/q} μ(ds, dz).

Here q is a parameter which controls whether jumps are classified as small jumps or big jumps. If the parameter q becomes larger, then less jumps are classified as small jumps and more jumps are classified as large jumps. Here, the process N (q) has almost surely only finitely many jumps in each compact time interval [0, t], t ≥ 0; compare Lemma 2.1.7 a) in [30]. A lot of results are easier to prove for

1 Framework

7

processes of finite jump activity. Therefore a common strategy in the upcoming chapters will be to first prove the results where we only consider N (q) instead of M (q) + N (q) and then in a second step to show that the asymptotic contribution of M (q) vanishes as q → ∞. A key observation which is especially important when disentangling asymptotic contributions of the continuous part of X and its jump part is that moments of the increments of B(q), C, M (q), N (q) scale differently with the corresponding interval length. Their specific behaviour is summarized in the following lemma which will be used repeatedly in the upcoming proofs. Lemma 1.4. If Condition 1.3 is fulfilled and the processes bt , σt and Γt are  p,q , eq ≥ 0 such that bounded there exist constants Kp , Kp , Kp,q , K B(q)s+t − B(q)s p ≤ Kp,q tp ,





p

p 2

E Cs+t − Cs  Fs ≤ Kp t ,  

p p E M (q)s+t − M (q)s p Fs ≤ Kp t 2 ∧1 (eq ) 2 ∧1 ,

    p ,q t + Kp ,q tp , E N (q)s+t − N (q)s p Fs ≤ K  

p E Xs+t − Xs p Fs ≤ Kp t 2 ∧1 ,

(1.8) (1.9) (1.10) (1.11) (1.12)

for all s, t ≥ 0 with s + t ≤ T and all q > 0, p ≥ 0, p ≥ 1. Here, eq can be  p ,q may be chosen chosen such that eq → 0 for q → ∞. For p ≥ 2 the constant K independently of q. Throughout the proofs in this book to simplify notation we denote by K and Ka generic constants, the latter dependent on some variable a. This means that e.g. statements like K = 2K or K = K 2 may occur. In fact the numeric value of these constants will never be of importance. The proof of Lemma 1.4 will be given in Appendix A. Lemma 1.4 can be used to (l) bound moments of increments over the observation intervals Ii,n if the observation scheme is deterministic and also if the observation times are exogenous. In the second case we can work conditionally on S and then apply the inequalities above for conditional expectations with respect to the σ-algebra σ(Fs , S). If we consider endogenous observation times we require more general results. These are stated in the following Lemma which will be proved in Appendix A as well. Lemma 1.5. If Condition 1.3 is fulfilled and the processes bt , σt and Γt are bounded there exist constants K, Kp , Kp,q such that sup t∈(S,S  ]

E

B(q)t − B(q)S p ≤ Kp,q (S − S)p ,

sup t∈(S,S  ]





Ct − CS p FS ≤ Kp E[(S − S)p/2 |FS ],

(1.13) (1.14)

8

1 Framework

E E



sup t∈(S,S  ]

sup t∈(S,S  ]





M (q)t − M (q)S 2 FS ≤ Keq E[(S − S)|FS ],





Xt − XS 2 FS ≤ K E[(S − S)|FS ],

(1.15) (1.16)

for all stopping times 0 ≤ S ≤ S and all q > 0, p ≥ 1. Here, eq can be chosen such that eq → 0 for q → ∞. Throughout the proofs in the upcoming chapters we will without further notice (1) (2) assume that the processes bt , σt , σt , ρt and Γt are bounded on [0, T ]. This assumption e.g. allows to directly apply the above lemmata. The processes bt , (1) (2) σt , σt , ρt and Γt are all locally bounded by Condition 1.3 and a localization procedure then shows that the results for bounded processes can be carried over to the case of locally bounded processes. See Section 4.4.1 in [30] for a detailed proof of the validity of this argument.

2 Laws of Large Numbers In the setting of high-frequency statistics for stochastic processes, the information (l) (l) contained in the observed data {X (l) : ti,n ≤ T }, l = 1, 2, is also (almost) ti,n

(l)

(l)

equivalently stored in the increments {Δi,n X (l) : ti,n ≤ T }, l = 1, 2. Note that to fully recover the observed data from the increments additionally only the starting (l) values X (l) , l = 1, 2, would be needed. However in many applications like volatility t0,n

estimation or the examination of the jump behaviour the properties of interest are invariant under constant shifts of the processes X (l) , l = 1, 2. Hence the (l) information contained in the starting values X (l) , l = 1, 2, is often not relevant t0,n

and it is sufficient to work with statistics which are based on the increments (l) (l) {Δi,n X (l) : ti,n ≤ T }, l = 1, 2. A very prominent class of such statistics is given by sums of certain transformations of the increments. Suppose for the moment that the observation scheme is synchronous and denote (1)

(2)

ti,n := ti,n = ti,n ,

Δi,n X := Xti,n − Xti−1,n .

(2.1)

It is a classic result in stochastic analysis, compare e.g. Theorem 23 in [41], that



P

Δi,n X (k) Δi,n X (l) −→ [X (k) , X (l) ]T ,

k, l ∈ {1, 2}.

(2.2)

i:ti,n ≤T

As a generalization of (2.2) the asymptotics of statistics of the form



f (Δi,n X)

(2.3)

i:ti,n ≤T

for various functions f : R2 → R have been investigated; compare e.g. Chapter 3 of [30]. Applications of such statistics include the construction of tests for the presence of (common) jumps in [2] and [32]. In this chapter, we discuss how statistics similar to (2.3) can be generalized to the setting of asynchronous observation times. Further we investigate the asymptotics of these generalized statistics and we will develop (weak) laws of large numbers for them if possible. We will compare those results to the results obtained in the setting of synchronous observation times, learn where the results differ and what © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 O. Martin, High-Frequency Statistics with Asynchronous and Irregular Data, Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics, https://doi.org/10.1007/978-3-658-28418-3_2

10

2 Laws of Large Numbers

kind of extra assumptions have to be made in the case of asynchronous observations to obtain results which match those in the setting of synchronous observation times. In Section 2.1 we investigate the asymptotics of non-normalized functionals, i.e. statistics of the form (2.3), andin Section 2.2 we discuss normalized functionals, i.e. statistics of the form np−1 ti,n ≤T f (Δi,n X) for some p ≥ 0 which is related to f . While the limits of the non-normalized functionals usually only depend on the jump part of X the limits of normalized functionals usually only depend on the continuous part of X. In Sections 2.3 and 2.4 we will discuss further extensions of the non-normalized and normalized functionals discussed in Sections 2.1 and 2.2.

2.1 Non-Normalized functionals First note that when considering functionals of the form (2.3) in the setting of asynchronous observation times it is not straightforward anymore for which pairs (1) (2) of increments (Δi,n X (1) , Δj,n X (2) ) the evaluation of the function f should be (1)

(2)

included in the sum. An idea utilized by [22] is to include f (Δi,n X (1) , Δj,n X (2) ) (1)

(2)

if and only if the observation intervals Ii,n , Ij,n overlap. In this case a consistent

estimator for the quadratic covariation of [X (1) , X (2) ]T is obtained by using the function f (x1 , x2 ) = x1 x2 also in the setting of asynchronous observations, i.e. they showed



(1)

(1) (2) i,j:ti,n ∨tj,n ≤T

P

(2)

Δi,n X (1) Δj,n X (2) 1{I (1) ∩I (2) =∅} −→ [X (1) , X (2) ]T i,n

(2.4)

j,n

in the case of a continuous Itˆ o semimartingale X and for exogenous observation times. A corresponding result which also holds for endogenous observations was later given in [21] and the extension to processes including jumps has been derived in [8]. The structure of the sum is illustrated in Figure 2.1. In the style of this famous Hayashi-Yoshida estimator for the quadratic covariation we define V (f, πn )T =



(1)

(2)

(1)

(2)

f (Δi,n X (1) , Δj,n X (2) )1{I (1) ∩I (2) =∅} i,n

i,j:ti,n ∨tj,n ≤T

j,n

for functions f : R2 → R. We will see that these functionals converge to similar limits as the functionals (2.3) in the setting of synchronous observation times for a large class of functions f , and not only for f (x1 , x2 ) = x1 x2 as in the case of the Hayashi-Yoshida estimator. Further we define V (l) (g, πn )T =



(l) i:ti,n ≤T

(l)

g(Δi,n X (l) ),

l = 1, 2,

2.1 Non-Normalized functionals

11

for functions g : R → R. We will also state an asymptotic result for V (l) (g, πn )T to compare the results in the setting of asynchronously observed bivariate processes to those in simpler settings and also because such a result will be used in Chapter 8. To describe the limits of the functionals V (f, πn )T and V (l) (g, πn )T we denote B(f )T =



(1)

(2)

f (ΔXs , ΔXs ),

s≤T



∗

B (f )T =

(1)

(2)

f (ΔXs , ΔXs )1{ΔX (1) ΔX (2) =0} , s

s≤T

B (l) (g)T =



(l)

g(ΔXs ),

s

(2.5)

l = 1, 2,

s≤T

for functions f : R2 → R and g : R → R for which the sums are well-defined.

2.1.1 The Results In the setting of synchronous observation times (2.1) the functional V (f, πn )T coincides with the classical statistic (2.3) and hence the convergence of V (f, πn )T in this situation follows from Theorem 3.3.1 of [30]. Theorem 2.1. Suppose that the observation scheme is synchronous and that Condition 1.3 holds. Then we have P

V (f, πn )T −→ B(f )T

(2.6)

for all continuous functions f : R2 → R with f (x) = o(x2 ) as x → 0. (1)

(1)

|I1,n |

(1)

|I2,n |

|I3,n |

(1)

(1)

|I4,n |

|I5,n |

X (1)  (1) (2)  I ∩I

1,n

1,n

 (1) (2)  I ∩I

2,n

1,n

 (1) (2)   (1) (2)  I ∩I I ∩I

2,n

2,n

3,n

3,n

 (1) (2)  I ∩I

4,n

3,n

 (1) (2)  I ∩I

5,n

3,n

X (2) (2)

|I1,n |

(2)

|I2,n |

(2)

|I3,n |

Figure 2.1: All products of increments of X (1) and X (2) over intersecting intervals enter the estimation of [X (1) , X (2) ]T .

12

2 Laws of Large Numbers

Actually the statement in Theorem 3.3.1 of [30] holds for general d-dimensional Itˆ o semimartingales and all functions f : Rd → R with the property f (x) = o(x2 ) for any d ∈ N. The case d = 1 then yields the convergence for the functionals V (l) (g, πn )T stated in the following corollary. Corollary 2.2. Under Condition 1.3 we have P

V (l) (g, πn )T −→ B (l) (g)T for all continuous functions g : R → R with g(x) = o(x2 ) as x → 0. The following theorem states the most general result which can be obtained if the convergence in (2.6) is supposed to hold for arbitrary Itˆ o semimartingales and any asynchronous observation scheme. Theorem 2.3. Under Condition 1.3 we have P

V (f, πn )T −→ B ∗ (f )T

(2.7)

for all continuous functions f : R2 → R with f (x, y) = O(x2 y 2 ) as |xy| → 0. As for the convergence in Theorem 2.1 in the setting of synchronous observation times we need that f (x, y) vanishes as (x, y) → 0 also in the setting of asynchronous observation times. However in the asynchronous setting we further need f (xk , yk ) → 0 also for sequences (xk , yk )k∈N which do not converge to zero, but which fulfil |xk yk | → 0. Hence the condition on f needed to obtain convergence of V (f, πn )T in the asynchronous setting is stronger compared to the corresponding condition in the synchronous setting. Further we observe that in the asynchronous setting the limit only consists of common jumps of X (1) and X (2) i.e. jumps with (1) (2) ΔXs = 0 = ΔXs . The following example illustrates the need for the stronger condition as well as why we only consider functions f which yield a limit that consists only of common jumps. Example 2.4. Consider the function f3,0 (x, y) = x3 , which fulfils f3,0 (x, y) → 0 (1)

as (x, y) → 0 but not as |xy| → 0, and the observation scheme given by ti,n = i/n (2)

and ti,n = i/(2n). Then V (f3,0 , πn )T = 2



(1)

P

(Δi,n X (1) )3 −→ 2B(f3,0 )T

(1)

i:ti,n ≤T

where the convergence is due to Corollary 2.2. However for the standard synchron(1)

(2)

P

ous observation scheme ti,n = ti,n = i/n we have V (f3,0 , πn )T −→ B(f3,0 )T also

2.1 Non-Normalized functionals

13

due to Corollary 2.2. Hence the limit here depends on the observation scheme. If (1) we further consider the observation scheme with ti,n = i/n and

 (2) ti,n

=

i/n,

n even,

i/(2n),

n odd,

then V (f3,0 , πn )T does not converge at all unless B(f3,0 )T = 0, as one subsequence converges to B(f3,0 )T and the other one to 2B(f3,0 )T . Hence there cannot exist a convergence result for V (f3,0 , πn )T which holds for any Itˆ o semimartingale X and any sequence of observation schemes πn , n ∈ N. If we consider instead a function f (x, y) that vanishes as |xy| → 0 such a behaviour cannot occur because idiosyncratic jumps do not contribute in the limit (1) (2) as e.g. for ΔXs = 0 and ΔXs = 0 we have sup (1) (1) (2) (i,j):s∈Ii,n ,Ii,n ∩Ij,n =∅

(1)

(2)

P

|Δi,n X (1) Δj,n X (2) | −→ 0.

If on the other hand there is a common jump at time s there is only one summand (1) (2) (1) (2) f (Δi,n X (1) , Δj,n X (2) ) such that s ∈ Ii,n and s ∈ Ij,n . Hence common jumps only enter the limit once. This example shows that the assumption that f (x, y) vanishes as |xy| → 0 is needed to filter out the contribution of idiosyncratic jumps. These jumps may enter V (f, πn )T multiple times, where the multiplicity by which they occur may depend on n and ω and therefore may prevent V (f, πn )T from converging.  Let us now consider the order by which the function f (x, y) has to decrease as (x, y) → 0 or, respectively, |xy| → 0. We observe that in the asynchronous setting the function f has to decrease quadratically in both x and y while in the synchronous setting it only has to decrease quadratically in (x, y). Adding this condition to the requirement that f (xk , yk ) has to vanish for any sequence with |xk yk | → 0 further diminishes the class of functions f for which V (f, πn )T converges in the asynchronous setting compared to the synchronous one. The need for this stronger condition on f is due to the fact that the lengths of the observation intervals of X (1) and X (2) may decrease with different rates in the asynchronous setting which is illustrated in the following example. (1)

Example 2.5. Let Xt = 1{t≥U } for U ∼ U [0, 1] and X (2) be a standard Brownian motion independent of U . The observation schemes are given by

14

2 Laws of Large Numbers nU −1 n

U

nU  n

X (1) 1/n X (2) n1+γ (nU −1)/n n1+γ

1/n

n1+γ nU /n n1+γ

1+γ

Figure 2.2: Observation times of X (1) and X (2) around the jump time U . (1)

(2)

ti,n = i/n and ti,n = i/n1+γ with γ > 0. Then for f (x, y) = |x|p1 |y|p2 as illustrated in Figure 2.2 we have n1+γ nU /n/n1+γ



V (f, πn )1 =

 (2) p 2 X 1+γ − X (2)  i/n (i−1)/n1+γ

i=n1+γ (nU −1)/n/n1+γ +1 nγ 

≥

  −(1+γ)/2 n p2 n Zi  i=1

=n

−(1+γ)p2 /2+γ

 n

−γ

  n p 2   Zi 

nγ  i=1

(2)

where Zin := n(1+γ)/2 Δn1+γ (nU −1)/n/n1+γ +i/n1+γ ,n X (2) , i = 1, . . . nγ , are i.i.d. standard normal random variables for each n ∈ N. Hence V (f, πn )1 diverges for p2 < 2 if γ>

p2 2 − p2

because the expression in parantheses converges in probability to E[|Z|p2 ] for some Z ∼ N (0, 1), by the law of large numbers. Here we are able to find a suitably large γ explicitly because the p2 -variations of a Brownian motion are infinite for p2 < 2. But we also have B ∗ (f )1 = 0 in this setting because X (2) is continuous. Hence (2.7) cannot hold for f (x, y) = |x|p1 |y|p2 , any Itˆ o semimartingale of the form (1.1) and any observation scheme which fulfils Condition 1.3 if p1 ∧ p2 < 2.  Example 2.5 shows that the convergence (2.7) fails for functions f (x, y) = |x|p1 |y|p2 with p1 ∧ p2 < 2 in combination with observation schemes where the observation frequency for one process increases much faster as n → ∞ than the observation

2.1 Non-Normalized functionals

15

frequency of the other process. If we consider only observation schemes where such a behaviour is prohibited, we can also obtain the convergence in (2.7) for functions f (x, y) = |x|p1 |y|p2 with p1 ∧ p2 < 2. First, we state a result in the case of exogenous observation times introduced in Definition 1.2, i.e. random observation times that do not depend on the process X or its components. Theorem 2.6. Assume that Condition 1.3 holds and that the observation scheme is exogenous. Further let p1 , p2 > 0 with p1 + p2 ≥ 2. If we have



 (1)  p1 ∧1  (2)  p2 ∧1 I  2 I  2 1 i,n

j,n

(1) (2) i,j:ti,n ∨tj,n ≤T

(1)

(2)

{Ii,n ∩Ij,n =∅}

= OP (1)

(2.8)

as n → ∞ it holds P

V (f, πn )T −→ B ∗ (f )T

(2.9)

for all continuous functions f : R2 → R with f (x, y) = o(|x|p1 |y|p2 ) as |xy| → 0. In the boundary case p1 + p2 = 2 in Theorem 2.6 we achieve convergence for all functions f that are for |xy| → 0 dominated by the function |x|p1 |y|p2 which is of order p1 + p2 = 2. Hence Theorem 2.6 allows to achieve the convergence in (2.9) for functions f which are dominated by functions of the same order as the dominating function (x, y)2 in the synchronous case in Theorem 2.1. However, the requirement that f (x, y) vanishes as |xy| → 0 cannot be relaxed because this assumption is as illustrated in Example 2.4 fundamentally necessary due to the asynchronous nature of the observation scheme. Like in the synchronous setting we cannot have the general convergence in (2.9) for functions f which do not fulfil f (x, y) = O((x, y)2 ) as (x, y) → 0, because in this case B ∗ (f ) might not be well defined. An indication for this fact is also given by the observation that (2.8) can never be fulfilled if p1 + p2 < 2 and |πn |T → 0 as shown in the following remark. Remark 2.7. Suppose that we have p1 +p2 < 2 and |πn |T → 0. In this situation we obtain the following estimate for the left-hand side of (2.8), using (pl /2) ∧ 1 = pl /2, N N p p l = 1, 2, and the inequality i=1 ai ≥ ( i=1 ai ) , which holds for all N ∈ N, ai ≥ 0, p ∈ [0, 1):



(1)

|Ii,n |

p1 2

∧1

(1) (2) i,j:ti,n ∨tj,n ≤T

≥

 (1)

i:ti,n ≤T

(1)

|Ii,n |

p1 2



(2)

|Ij,n |

p2 2

 (2)

j:tj,n ≤T

∧1

1{I (1) ∩I (2) =∅} i,n

j,n

(2)

|Ij,n |1{I (1) ∩I (2) =∅} i,n

j,n

 p22

16

2 Laws of Large Numbers ≥



(1)

p1 2

p1 +p2 2

−1

|Ii,n |

(1)

|Ii,n |

p2 2

− O((|πn |T )

p1 +p2 2

)

(1) i:ti,n ≤T

≥ (|πn |T )

T − O((|πn |T )

p1 +p2 2

).

Here the expression in the last line converges in probability to infinity due to P

p1 + p2 < 2 and |πn |T −→ 0. On the other hand the condition (2.8) is always fulfilled if p1 ∧ p2 ≥ 2 holds. Indeed in that case we obtain

 (1)

(1)

|Ii,n |

p1 2

∧1

(2)

|Ij,n |

p2 2

(2)

i,j:ti,n ∨tj,n ≤T



≤K

(1)

(2)

∧1

1{I (1) ∩I (2) =∅} i,n

(1)

j,n

(2)

|Ii,n ||Ij,n |1{I (1) ∩I (2) =∅} ≤ 3K|πn |T T.

i,j:ti,n ∨tj,n ≤T

i,n

j,n

Hence condition (2.8) is by Condition 1.3 always fulfilled in the setting of Theorem 2.3.  Example 2.8. Let p ∈ [1, 2) and consider the deterministic sampling scheme (1) (2) given by ti,n = i/n and ti,n = i/n1+γ from Example 2.5 with γ = γ(p) = 2p−2 2−p . In this case it holds   p2   (1)  (2)  p2 1 1+γ(p) 1 I I  1 (1) (2) = T n (1 + o(1)) i,n j,n {Ii,n ∩Ij,n =∅} n n1+γ(p) (1) (2) i,j:ti,n ∨tj,n ≤T

p

= T n1+γ(p)−(2+γ(p)) 2 (1 + o(1)) = O(1).

Hence if we want (2.9) to hold for all functions f with f (x, y) = o(|xy|p ) we can allow for observation schemes where the observation frequencies differ by a factor of up to nγ(p) where γ(p) increases in p. For p = 1 we have γ(1) = 0 and for p → 2 we have γ(p) → ∞.  In general we observe that if the o(|x|p1 |y|p2 )-restriction on f is less restrictive, then the restriction (2.8) on the observation scheme has to be more restrictive, and vice versa. Here the abstract criterion (2.8) characterizing the allowed classes of observation schemes can be related, as illustrated in Example 2.8, to the asymptotics of the ratio of the observation frequencies of the two processes. Hence if the observation frequency of one process increases much faster than the observation frequency of the other process we obtain the convergence in (2.9) only for a small class of functions f . Example 2.9. Condition (2.8) is fulfilled for any p1 , p2 ≥ 0 with p1 + p2 = 1 in (l) the case where the observation times {ti,n : i ∈ N}, l = 1, 2, are given by the jump

2.1 Non-Normalized functionals

17

times of two independent time-homogeneous Poisson processes with intensities nλ1 , nλ2 . Indeed in that situation Corollary 5.6 yields



 (1)  p1  (2)  p2 I  2 I  2 1 i,n

(1)

j,n

(2)

i,j:ti,n ∨tj,n ≤t

(1)

(2)

{Ii,n ∩Ij,n =∅}

P

−→ ct, t ≥ 0,

for some positive real number c > 0. Further note that if (2.8) is fulfilled for p1 , p2 ≥ 0 it is clearly also fulfilled for any p 1 ≥ p1 , p 2 ≥ p2 .  Next, we give a result that may also be applied in a setting with endogenous observation times. Theorem 2.10. Assume that Condition 1.3 holds. If for all ε > 0 there exists some Nε ∈ N with lim sup P n→∞



sup



(l)

i:ti,n ≤T j∈N

 1{I (l) ∩I (3−l) =∅} > Nε , l = 1, 2 < ε, i,n

(2.10)

j,n

then it holds P

V (f, πn )T −→ B ∗ (f )T for all continuous functions f : R2 → R such that f (x, y) = o(|x|p1 |y|p2 ) as |xy| → 0 for some p1 , p2 ≥ 0 with p1 + p2 = 2. Here (2.10) ensures that as n tends to infinity the maximal number of observations of the process X (3−l) during one observation interval of X (l) is bounded. This yields that the ratio of the observation frequencies of the two processes is also bounded as n → ∞ and cannot tend to infinity as in Example 2.5. Example 2.11. Consider the case where X (1) and X (2) are observed alternately. In this case we have



j∈N

1{I (l) ∩I (3−l) } ≤ 2 i,n

j,n

for l = 1, 2 and all i, n. Note that, although this goes along with a data reduction, the statistician may always use only a subset of all available observations and hence is able to turn the real observation scheme for example into an obervation scheme where the processes are observed alternately. One way to achieve this is the following: Start with (1) (1) (1) the first observation time ti,n of X (1) and set t˜1,n = t1,n , then take the smallest

(1) (2) (2) (2) (1) observation time of X (2) larger than t˜i,n and set t˜1,n = inf{ti,n |ti,n > t˜1,n }. (1) (1) (1) (2) Further set t˜2,n = inf{ti,n |ti,n > t˜1,n } and define recursively the new observation scheme π ˜n by continuing this procedure in the natural way. 

18

2 Laws of Large Numbers

Remark 2.12. It can be shown that Theorems 2.1, 2.3, 2.6 and 2.10 do not just hold for continuous functions f : R2 → R but also for functions f : R2 → R which are only discontinuous at points (x, y) ∈ R2 for which almost surely no jump ΔXs is realized in [0, T ] with ΔXs = (x, y). Hence if the jump measure admits a Lebesgue-density the convergences in Theorems 2.1, 2.3, 2.6 and 2.10 remain valid for all functions f where the set of all discontinuity points is a Lebesgue null set. For a more precise formulation of the above claim see Theorem 3.3.5 in [30]. In this section, we restricted ourselves to the case of continuous f because all functions for which these results will be applied in Part II are continuous and to keep the the notation and the proofs clearer. 

2.1.2 The Proofs As a preparation for the proof of Theorem 2.3 we prove (2.7) for functions f that vanish in a neighbourhood of the two axes {(x, y) ∈ R2 |xy = 0} in Lemma 2.13 and for the function f ∗ : R2 → R, (x, y) → x2 y 2 in Lemma 2.14. Lemma 2.13. Under Condition 1.3 we have P

V (f, πn )T −→ B ∗ (f )T for all continuous functions f : R2 → R for which some ρ > 0 exists such that f (x, y) = 0 whenever (x , y ) ∈ {(x, y) ∈ R2 : |x y | < ρ}. Proof. The following arguments are similar to the proof of Lemma 3.3.7 in [30]: Note that as X is c` adl` ag there can only exist countably many jump times s ≥ 0 (1) (2) with |ΔXs ΔXs | ≥ ρ/2 and in each compact time interval there are only finitely many such jumps. Denote by (Sp )p∈N an enumeration of those jump times and let

 t = Xt − X

 t 0

R2

δ(s, z)1{|δ(1) (s,z)δ(2) (s,z)|≥ρ/2} μ(ds, dz) (1)

(2)

 s ΔX s | < ρ/2 for denote the process X without those jumps. This yields |ΔX all s ∈ [0, T ]. Hence (1)

lim sup

sup

θ→0

0≤sl ≤tl ≤T −|πn |T ,tl −sl ≤θ,(s1 ,t1 ]∩(s2 ,t2 ] =∅

(1)

(2)

(2)

 (ω) − X s1 (ω))(X  (ω) − X s2 (ω))| < |(X t1 t2

ρ 2

for all ω ∈ Ω. Then there exists θ : Ω → (0, ∞) such that (1)

sup 0≤sl ≤tl ≤T,tl −sl

≤θ  (ω),(s

1 ,t1 ]

∩(s2 ,t2 ] =∅

(1)

(2)

(2)

 (ω) − X s1 (ω))(X  (ω) − X s2 (ω))| < ρ. |(X t1 t2

Denote by Ω(n) the subset of Ω which is defined as the intersetion of the set {|πn |T ≤ θ } and the set on which any two different jump times Sp = Sp with

2.1 Non-Normalized functionals

19

Sp , Sp ≤ T satisfy |Sp − Sp | > 2|πn |T and on which |T − Sp | > |πn |T for any Sp ≤ T . Then we have



V (f, πn )T 1Ω(n) =



f Δ

p:Sp ≤T −|πn |T

(1) (1) in (Sp ),n

X (1) , Δ



(2) (2) in (Sp ),n

X (2) 1Ω(n) (2.11)

(l)

where in (s) denotes the index of the interval characterized by s ∈ I Further we get from Condition 1.3



f Δ

(1) (1) in (Sp ),n

X (1) , Δ



(2) (2) in (Sp ),n



P

(l) (l)

in (s),n

.

(2) 

(1)

X (2) 1{Sp ≤T } −→ f ΔXSp , ΔXSp 1{Sp ≤T }

for any p ∈ N because X is c` adl` ag and f is continuous. Using this convergence, the fact that there exist almost surely only finitely many p ∈ N with Sp ≤ T and P(ΔXT = 0) = 1 we obtain





f Δ

p:Sp ≤T −|πn |T

(1) (1) in (Sp ),n

P

−→

X (1) , Δ

 

(2) (2) in (Sp ),n

X (2)

(2) 

(1)

f ΔXs , ΔXs



1{|ΔX (1) ΔX (2) |≥ρ/2} = B ∗ (f )T , s

s≤T

s

where the last equality holds because of f (x, y) = 0 for |xy| < ρ. This yields the claim because of (2.11) and P(Ω(n)) → 1 as n → ∞. Lemma 2.14. Under Condition 1.3 we have P

V (f ∗ , πn )T −→ B ∗ (f ∗ )T

(2.12)

for f ∗ : R2 → R, (x, y) → x2 y 2 . Proof. The convergence (2.12) follows from 

lim lim sup P 

q→∞ n→∞





(1) (2) i,j:ti,n ∨tj,n ≤T

(1)

(2)



2



Δi,n N (1) (q)Δj,n N (2) (q) 1{I (1) ∩I (2) =∅} − B ∗ (f ∗ )T  > ε → 0 i,n

j,n

(2.13) and 

lim lim sup P V (f ∗ , πn )T −

q→∞ n→∞

 (1) (2) i,j:ti,n ∨tj,n ≤T



(1)

(2)



2



Δi,n N (1) (q)Δj,n N (2) (q) 1{I (1) ∩I (2) =∅}  > ε → 0 i,n

j,n

(2.14) for any ε > 0. We will prove (2.13) and (2.14) in the following.

20

2 Laws of Large Numbers

For proving (2.13) we denote by Ω(n, q) the set on which two different jumps of N (q) are further apart than 2|πn |T . On Ω(n, q) we have

 (1)



(1)

i,n

(2)

i,j:ti,n ∨tj,n ≤T

=

2

(2)

Δi,n N (1) (q)Δj,n N (2) (q) 1{I (1) ∩I (2) =∅} 1Ω(n,q)



ΔN (1) (q)s

2 

ΔN (2) (q)s

2

j,n

1Ω(n,q) .

(2.15)

s≤T

Note that the sum without the indicator in (2.15) converges to BT (f ∗ ) for q → ∞. Thus, (2.13) follows since P(Ω(n, q)) → 1 as n → ∞ for any q > 0. For proving (2.14) we apply inequality (2.46) for the function f (x1 , x2 ) = x1 2 x2 2 and (1)

(1)

(1)

(1)

(2)

(2)

(2)

(2)

x1 = Δi,n N (1) (q), y1 = Δi,n B (1) (q) + Δi,n C (1) + Δi,n M (1) (q), x2 = Δj,n N (2) (q), y2 = Δj,n B (2) (q) + Δj,n C (2) + Δj,n M (2) (q) which for arbitrary ε > 0 yields



 V (f ∗ , πn )T −

(1)



(1)

i,n

(2)

i,j:ti,n ∨tj,n ≤T



≤ θ(ε )

(1)



(2)

(1)

×







l=1,2 i,j:t(l) ∨t(3−l) ≤T j,n

(3−l)

Δj,n B (3−l) (q)

+ Kε 

2

(2)

i,n



i,n

2 



(l)

Δi,n B (l) (q) (l)

+ Δj,n C (3−l) (3−l)





2

2

+ Δj,n N (3−l) (q)



j,n

Δi,n N (1) (q)Δj,n N (2) (q) 1{I (1) ∩I (2) =∅}

i,j:ti,n ∨tj,n ≤T

+ Kε 



2

(2)

Δi,n N (1) (q)Δj,n N (2) (q) 1{I (1) ∩I (2) =∅} 



(l)

+ Δi,n M (l) (q)



(2.16)

j,n

(3−l)

2 

+ Δj,n M (l) (q)

1{I (l) ∩I (3−l) =∅} i,n

j,n

2

2 

(2.17)

2  (3−l) (3−l) 2 (l) Δi,n C (l) Δj,n C

+



2  (l) Δj,n N (3−l) (q)

l=1,2 i,j:t(3−l) ∨t(l) ≤T i,n j,n

× 1{I (l) ∩I (3−l) =∅} . i,n

(2.18)

j,n

First note that for (2.16) we obtain 

lim lim lim sup P θ(ε ) 

ε →0 q→0 n→∞

 (1)

(2)

i,j:ti,n ∨tj,n ≤T



(1)

(2)

2



Δi,n N (1) (q)Δj,n N (2) (q) 1{I (1) ∩I (2) =∅} > ε = 0 i,n

j,n

2.1 Non-Normalized functionals

21

for any ε > 0 using (2.13). Furthermore for (2.17) it holds



Kε 

(l)



(l)

Δi,n B (l) (q)

(3−l)

+ Δj,n C (3−l)

2

(l)



(3−l)

+ Δj,n M (3−l) (q)

× 1{I (l) ∩I (3−l) =∅} i,n

≤ Kε 



+ Δi,n M (l) (q)

2  

(3−l)

Δj,n B (3−l) (q)

2

(l)

i,j:ti,n ∨tj,n ≤T



2



j,n





(l)

Δi,n B (l) (q)

2



2



(3−l)

+ Δj,n N (3−l) (q)

(l)

+ Δi,n M (l) (q)

2 

2 

(l)

×



i:ti,n ≤T

 (3−l)

j:tj,n



P



(3−l)

Δj,n B (3−l) (q)

≤T



(3−l)

+ Δj,n M (3−l) (q)

2

2



(3−l)

+ Δj,n C (3−l)



(3−l)

2

+ Δj,n N (3−l) (q)

−→ Kε [B (l) (q), B (l) (q)]T + [M (l) (q), M (l) (q)]T



2 

X (3−l) , X (3−l)

 T

which tends to zero for q → ∞ as [B (l) (q), B (l) (q)]T = 0 for any q > 0. For the treatment of the remaining terms (2.18) we set 

Kn,T (l, ρ) =

m  

sup (l) (l) (l) m,m ∈N,m≤m ,|tm ,n −tm−1,n |≤ρ,tm ,n ≤T

k=m

C

(l) (l)

tk,n

−C

2

(l) (l)

tk−1,n

,

l = 1, 2. Then we have lim lim sup P(Kn,T (l, ρ) > δ) = 0

(2.19)

ρ→0 n→∞

for any δ > 0 due to the u.c.p. convergence of realized volatility to the quadratic variation; see Theorem II.22 in [41]. In fact on the set {Kn,T (l, ρ) > δ} it holds ∞   (l)  C (l) sup 

0≤s≤T

i=1

ti,n ∧s

−C

2

(l) (l) ti−1,n ∧s



−

s 0

 

(l)

(σu )2 du1{Kn,T (l,ρ)>δ} ≥

δ − K2ρ 2

where σs  ≤ K. This is illustrated in Figure 2.3 and yields (2.19). Using the fact that the total length of the observation intervals of one process which overlap with a specific observation interval of the other process is at most 3|πn |T , we get on the set {3|πn |T ≤ ρ}



Kε (l)

(3−l)

i,j:ti,n ∨tj,n

(l)

(l)



(3−l)

(Δi,n C (l) )2 (Δj,n C (3−l) )2 + Δj,n N (3−l) (q)

2 

≤T

× 1{I (l) ∩I (3−l) =∅} 1{3|πn |T ≤ρ} i,n

≤ K Kn,T (l, ρ)

j,n



ε

(3−l)

j:tj,n

≤T



(3−l)

(3−l)



(Δj,n C (3−l) )2 + (Δj,n N (3−l) (q))2 1{3|πn |T ≤ρ} .

22

2 Laws of Large Numbers

s →

b s → δ≤

∞  i=1

s 0

C

(l) (l)

ti,n ∧s

−C

2

(l) (l)

ti−1,n ∧s

(l)

(σu )2 du

≤ K2ρ a

bla

(l)

(l)

tm ,n

tm−1,n

s ≤ρ

Figure 2.3: Either a or b has to be larger than (δ − K 2 ρ)/2. As the latter sum converges to the quadratic variation of X (l) as first n → ∞ and then q → ∞, we obtain that the sum (2.18) vanishes by (2.19) and because of P(3|πn |T ≤ ρ) → 1 as n → ∞ for any fixed ρ > 0. Proof of Theorem 2.3. Define fρ (x, y) = f (x, y)ψ(|xy|/ρ) where ψ : [0, ∞) → [0, 1] is continuous with ψ(u) = 0 for u ≤ 1/2 and ψ(u) = 1 for u ≥ 1. By Lemma 2.13 we have P

V (fρ , πn )T −→ B ∗ (fρ )T

(2.20)

for all ρ > 0. Because of f (x, y) = O(x2 y 2 ) as |xy| → 0 there exist constants Kρ > 0 with |fρ (x, y)| ≤ |f (x, y)| ≤ Kρ |x|2 |y|2 for all (x, y) with |xy| < ρ and ρ → Kρ is non-increasing as ρ → 0. Hence it holds |B ∗ (f )T − B ∗ (fρ )T | ≤ 2





 



(1) 2 (2) 2 Kρ ΔXs  ΔXs  1{|ΔX (1) ΔX (2) | δ) = 0

ρ→0 n→∞

(2.22)

2.1 Non-Normalized functionals

23

for any δ > 0 in addition to (2.20) and (2.21). To this end we consider the following inequality which is obtained like (2.21)



|V (f, πn )T − V (fρ , πn )T | ≤

(1)



 



2 2 (1) (2) 2Kρ Δi,n X (1)  Δj,n X (2) 

(2)

i,j:ti,n ∨tj,n ≤T

× 1{|Δ(1) X (1) Δ(2) X (2) | 0. Denote by Sq,p the jump times of N (q) and let Ω(n, q, ρ) be the set on which no two jump times Sq,p , Sq,p ≤ T fulfil |Sq,p − Sq,p | ≤ 2|πn |T , it holds Sq,p ≤ T − |πn |T and |ΔN (1) (q)Sq,p ΔN (2) (q)Sq,p − Δi(1) (S n

q,p ),n

X (1) Δi(2) (S n

q,p p),n

X (2) | ≤ ρ

ρ the jump times of N (q) with for any Sq,p ≤ T . Further we denote by Sq,p

|ΔN (1) (q)Sρ ΔN (2) (q)Sρ | < 2ρ. q,p

q,p

Then it holds



(1)



 



2 2 (1) (2) 2Kρ Δi,n N (1) (q) Δj,n N (2) (q)

(2)

i,j:ti,n ∨tj,n ≤T

≤ 2Kρ

 ρ q,p p:S ≤T

× 1{|Δ(1) X (1) Δ(2) X (2) | 0; compare Theorem 6.7 in [33]. Proof of Theorem 2.6. Set fρ (x, y) = f (x, y)ψ(|xy|/ρ) like in the proof of Theorem P

2.3. As in the proof of Theorem 2.3 we obtain V (fρ , πn )T −→ B ∗ (fρ )T and



lim |B ∗ (f )T − B ∗ (fρ )T | ≤ lim

ρ→0

ρ→0

≤ lim Kρ ρ→0



(1)

(2)

2Kρ |ΔXs |p1 |ΔXs |p2 1{|ΔX (1) ΔX (2) | 0.

Because of f (x, y) = o(|x|p1 |y|p2 ) as |xy| → 0 we obtain as in (2.23)   (1) (1) p1  (2) (2) p2 Δ X  Δ X  1 |V (f, πn )T − V (fρ , πn )T | ≤ 2Kρ i,n

(1)

(2)

i,j:ti,n ∨tj,n ≤T

j,n

(2.26)

(1)

(2)

{Ii,n ∩Ij,n =∅}

(2.27)

2.1 Non-Normalized functionals

25

with Kρ → 0 as ρ → 0. Define stopping times (Tkn )k∈N0 via T0n = 0 and (l)

n |i ∈ N0 , l = 1, 2}, Tkn = inf{ti,n > Tk−1

k ≥ 1.

Hence the Tkn mark the times, where at least one of the processes X (1) or X (2) is newly observed. Further we set (l)

(l)

τn,− (s) = sup{ti,n ≤ s|i ∈ N0 }, (l)

(2.28)

(l)

τn,+ (s) = inf{ti,n ≥ s|i ∈ N0 }

for the observation times of X (l) , l = 1, 2, immediately before and after s. Then we denote (l)

(l)

(l) Δn = XTk − XTk−1 , kX (l)

X (l) = XTk−1 − X Δn,l,− k X (l) = X Δn,l,+ k

(l)

(l) (l)

n τn,− (Tk−1 )

,

(2.29)

(l)

(l)

τn,+ (Tkn )

− X Tk

for l = 1, 2. Using this notation we obtain



 (1) (1) p1  (2) (2) p2 Δ X  Δ X  1 i,n

j,n

(1) (2) i,j:ti,n ∨tj,n ≤T



≤

(1)

(2)

{Ii,n ∩Ij,n =∅}

2   n,l,− (l) p (l) Δ X + Δn + Δn,l,+ X (l)  l kX k

k

k:Tkn ≤T l=1



≤

2 











 

p p (l) pl Kpl Δn,l,− X (l)  l + Δn + Δn,l,+ X (l)  l . kX k k

k:Tkn ≤T l=1

The S-conditional expectation of this quantity is bounded by

 k:Tkn ≤T

E[

2  l=1

= Kp 1 Kp 2



k:Tkn ≤T

+ Kp 1 Kp 2







n,l,+ (l) pl (l) pl Kpl |Δn,l,− X (l) |pl + Δn X | |S] k X | + |Δk k



(1) p1 (2) p2 E[|Δn | |Δn | |S] kX kX



l=1,2 k:Tkn ≤T

E |Δn,l,− X (l) |pl k  

(3−l) p3−l n × E[|Δn | + |Δn,3−l,+ X (3−l) |p3−l |σ(FTk−1 , S)]S kX k

+ Kp 1 Kp 2





l=1,2 k:Tkn ≤T

 

n,3−l,+ (3−l) p3−l (l) pl E |Δn X | |σ(FTkn , S)]S k X | E[|Δk (2.30)

26

2 Laws of Large Numbers (1) Δn 2X

(1) Δn,1,− k−1 X

Δn,1,+ X (1) 2

(1) Δn kX

X (1) T0n = 0

T1n

T2n

n Tk−1

T3n

Tkn

X (2) (2) Δn,2,− X (2) Δn 1 2X

(2) Δn kX

Figure 2.4: Merged observation times and the corresponding intervals (l) Δn,k,− X (l) , Δn and Δn,k,+ X (l) . kX k k where we used |Δn,1,− X (1) |p1 |Δn,2,− X (2) |p2 = |Δn,1,+ X (1) |p1 |Δn,2,+ X (2) |p2 = 0 k k k k which holds because one of each two increments is always zero. Further, using (1) p1 (2) p2 p1 +p2 |Δn | |Δn | ≤ 2Δn , kX kX k X

which we obtain as in (2.25), and inequality (1.12), (2.30) is bounded by Kp 1 Kp 2



p1 +p2 E[Δn |S] k X

k:Tkn ≤T

+ Kp 1 Kp 2





(3−l)

E[|Δn,l,− X (l) |pl K(τ+ k

n (Tkn ) − Tk−1 )

p3−l 2

∧1

|S]

l=1,2 k:Tkn ≤T





l=1,2

k:Tkn ≤T

+ Kp 1 Kp 2 ≤ Kp 1 Kp 2



(Tkn ) − Tkn )

p3−l 2

∧1

|S]

n K(Tkn − Tk−1 )

k:Tkn ≤T

+ 4Kp1 Kp2

(3−l)

(l) pl E[|Δn k X | K(τ+





k:Tkn ≤T l=1,2

≤ Kp1 ,p2 T + Kp1 ,p2

(l)

(l)

n (τ+ (Tkn ) − τ− (Tk−1 ))



(1) (2) i,j:ti,n ∨tj,n ≤T

(1)

K|Ii,n |

p1 2

∧1

pl 2

(2)

|Ij,n |

∧1

p2 2

∧1

1{I (1) ∩I (2) =∅} . i,n

j,n

This expression is bounded in probability by condition (2.8) and hence the righthand side of (2.27) vanishes for ρ → 0 due to Kρ → 0 as ρ → 0 which yields (2.26).

2.2 Normalized Functionals

27

Proof of Theorem 2.10. Comparing the proof of Theorem 2.6 it is sufficient to show that



Y (n) =

 (1) (1) p1  (2) (2) p2 Δ X   Δ X  1 i,n

(1)

j,n

(2)

i,j:ti,n ∨tj,n ≤T

(1)

(2)

{Ii,n ∩Ij,n =∅}

is bounded in probability as n → ∞. To this end fix ε > 0 and define



Ω(n, ε) = { sup (l)

i:ti,n ≤T j∈N

1{I (l) ∩I (3−l) =∅} > Nε , l = 1, 2} i,n

j,n

with Nε as in (2.10). We then obtain using Muirhead’s inequality like in (2.25) Y (n)1Ω(n,ε)c



≤



(1) (2) i,j:ti,n ∨tj,n ≤T

≤ Nε





(1)

Δi,n X (1)



(l)

2

Δi,n X (l)



(2)

+ Δj,n X (2)

2 

1{I (1) ∩I (2) =∅} 1Ω(n,ε)c i,n

j,n

2

l=1,2 i:t(l) ≤T i,n

where the sums in the last line converge to [X (l) , X (l) ]T , l = 1, 2. If we further choose Kε > 0 such that P([X (l) , X (l) ]T > Kε (1 − ξ)) ≤ ε, l = 1, 2, for some ξ ∈ (0, 1) we obtain lim sup P(Y (n) > 2Nε Kε ) n→∞

≤ lim sup P(Ω(n, ε)) + lim sup n→∞

n→∞

≤ ε + lim sup n→∞

+ lim sup n→∞

 l=1,2



 P 





P



l=1,2



(l)





(l)

Δi,n X (l)

2

> Kε



(l)

i:ti,n ≤T

Δi,n X (l)

2



− [X (l) , X (l) ]T  > Kε ξ



(l)

i:ti,n ≤T

 2  P [X (l) , X (l) ]T > Kε (1 − ξ)

l=1,2

≤ 3ε. As ε > 0 can be chosen arbitrarily this yields the boundedness in probability of Y (n), n ∈ N.

2.2 Normalized Functionals In Section 2.1 we have seen that the functional V (f, πn )T converges to a limit which depends only on the jump part of X for functions f that decay sufficiently fast in

28

2 Laws of Large Numbers

a neighbourhood of zero. This is necessary because we need that for such functions f the contribution of the continuous part in V (f, πn )T becomes asymptotically negligible. The jump part has the property that the magnitude of its increments remains constant as |πn |T → 0 while for the continuous martingale part

 Ct =

t 0

σs dWs , t ≥ 0, (l)

(l)

the magnitude of the normalized“ increment |Ii,n |−1/2 Δi,n C (l) remains constant. ” Hence if we would like to learn something about the continuous part of X it is (l) (l) reasonable to look at functionals of the normalized increments |Ii,n |−1/2 Δi,n X (l) . As an illustration for the upcoming results consider the toy example Xttoy = σWt

(2.31)

where the volatility matrix

 σ=

σ (1) ρσ (2)



0 1 − ρ2 σ (2)



is constant in time with σ (1) , σ (2) > 0 and ρ ∈ [−1, 1]. Suppose the observation scheme is exogenous, compare Definition 1.2, and synchronous. Under Condition 1.3 we then obtain using the notation from (2.1)





|Ii,n |f (|Ii,n |−1/2 Δi,n X toy ) =

i:ti,n ≤T

P

|Ii,n |f (σZin ) −→ T E[f (σZ)]

i:ti,n ≤T

(2.32) P

because of |πn |T −→ 0, where Z and Zin = |Ii,n |−1/2 Δi,n W , i ∈ N0 , are i.i.d. two-dimensional standard normal random variables for each n ∈ N. Functionals of this form are discussed in Section 14.2 of [30]. Two straightforward generalizations of this approach to the setting of asynchronous observation times lead to functionals of the form  (1) (2) i,j:ti,n ∨tj,n ≤T

(1)

(2)

(1)

(1)

(2)

(2)

(|Ii,n ||Ij,n |)1/2 f (|Ii,n |−1/2 Δi,n X (1) , |Ij,n |−1/2 Δj,n X (2) )1{I (1) ∩I (2) =∅} i,n

j,n

(2.33) and  (1) (2) i,j:ti,n ∨tj,n ≤T

(1)

(2)

(1)

(1)

(2)

(2)

|Ii,n ∩ Ij,n |f (|Ii,n |−1/2 Δi,n X (1) , |Ij,n |−1/2 Δj,n X (2) )1{I (1) ∩I (2) =∅} . i,n

j,n

2.2 Normalized Functionals

29

Here the main difference compared to the functional from (2.32) in the synchronous (1) (1) (2) (2) setting is that the law of (|Ii,n |−1/2 Δi,n X (1) , |Ij,n |−1/2 Δj,n X (2) ) is in general not independent of the observation scheme πn . This property is due to the fact that (1) (1) (2) (2) e.g. the correlation of |Ii,n |−1/2 Δi,n X toy,(1) and |Ij,n |−1/2 Δj,n X toy,(2) equals (1)

ρ

(2)

|Ii,n ∩ Ij,n | (1)

(2.34)

(2)

|Ii,n |1/2 |Ij,n |1/2

as the increments of X toy,(1) and X toy,(2) are correlated only over the overlapping (1) (2) part Ii,n ∩ Ij,n . This difference is also the main reason why it is more difficult to derive convergence results for normalized functionals as we will see later on. Further, regarding (2.33) the quantity

 (1)

(1)

(2)

(|Ii,n ||Ij,n |)1/2 1{I (1) ∩I (2) =∅}

(2)

i,n

i,j:ti,n ∨tj,n ≤T

j,n

might diverge for |πn |T → 0 as has been shown in Example 2.8. Due to these observations we will pick another approach where we use a global (l) (l) normalization instead of locally normalizing each Δi,n X (l) with |Ii,n |1/2 . Precisely, we have to assume that the average“ observation frequency increases with rate n. ” We then look at functionals of the form



(1)

(2)

(1)

(2)

n−1 f (n1/2 Δi,n X (1) , n1/2 Δj,n X (2) )1{I (1) ∩I (2) =∅} . (2.35) i,n

i,j:ti,n ∨tj,n ≤T

j,n

Such functionals also appear to occur more naturally in the applications discussed in Part II. The most common functions for which the functionals (2.35) are studied are the power functions gp (x) = xp , g p = |x|p and f(p1 ,p2 ) = xp11 xp22 , f (p1 ,p2 ) = |x1 |p1 |x2 |p2 where p, p1 , p2 ≥ 0. Those functions are members of the following more general classes of functions; compare Section 3.4.1 in [30]. Definition 2.16. A function f : Rd → R is called positively homogeneous of degree p ≥ 0, if f (λx) = λp f (x) for all x ∈ Rd and λ ≥ 0. Further f is called positively homogeneous with degree pi ≥ 0 in the i-th argument if the function x → f (x1 , . . . , xi−1 , x, xi+1 , . . . , xd ) is positively homogeneous of degree pi for any choice of (x1 , . . . , xi−1 , xi+1 , . . . , xd ) ∈ Rd−1 .  If the function f is positively homogeneous with degree p1 in the first argument and with degree p2 in the second argument, (2.35) becomes n(p1 +p2 )/2−1



(1)

(2)

i,j:ti,n ∨tj,n ≤T

(1)

(2)

f (Δi,n X (1) , Δj,n X (2) )1{I (1) ∩I (2) =∅} . i,n

j,n

30

2 Laws of Large Numbers

As we are going to derive results only for such functions we denote by



V (p, f, πn )T = np/2−1

(1)

(1)

(2)

f (Δi,n X (1) , Δj,n X (2) )1{I (1) ∩I (2) =∅} ,

(2)

i,n

i,j:ti,n ∨tj,n ≤T

j,n

(2.36) f : R2 → R, the functional whose asymptotics we are going to study in this section. Further we set V

(l)



(p, g, πn )T = np/2−1

(l)

g(Δi,n X (l) ), l = 1, 2,

(2.37)

(l) i:ti,n ≤T

for functions g : R → R. As in Section 2.1 we will also derive an asymptotic result (l)

for V (p, g, πn )T to compare the results in the setting of asynchronously observed bivariate processes to those in simpler settings. To describe the limits of the normalized functionals (2.36) and (2.37) in the upcoming results we need to introduce some notation. Denote by mΣ (h) = E[h(Z)], Z ∼ N (0, Σ),

(2.38)

the expectation of a function h : Rd → R evaluated at a d–dimensional centered normal distributed random variable with covariance matrix Σ. Further we define the expressions (l),n

Gp

  (l) p/2 I  ,

(t) = np/2−1

i,n

(l)

i:ti,n ≤t



(p1 +p2 )/2−1 Gn p1 ,p2 (t) = n

(1)

(1)

(2)

|Ii,n |p1 /2 |Ij,n |p2 /2 1{I (1) ∩I (2) =∅} ,

(2)

i,n

i,j:ti,n ∨tj,n ≤t

n (t) Hk,m,p

=n

p/2−1

 (1)

(1) |Ii,n

j,n

(2.39) (2) (2) \ Ij,n |k/2 |Ij,n

(1) \ Ii,n |m/2

(2)

i,j:ti,n ∨tj,n ≤t (1)

(2)

× |Ii,n ∩ Ij,n |(p−(k+m))/2 1{I (1) ∩I (2) =∅} , i,n

whose limits, if they exist, will occur in the limits of V

(l)

j,n

(p, g, πn )T and V (p, f, πn )T .

2.2.1 The Results We start with a result for V (p, f, πn )T under the restriction that the observations are synchronous as in (2.1). If we consider a function f : R2 → R which is positively homogeneous of degree p in our toy example (2.31) we get

E[V (p, f, πn )T |S] = np/2−1



i:ti,n ≤T

|Ii,n |p/2 E[f (|Ii,n |−1/2 Δi,n X toy )|S]

2.2 Normalized Functionals

31 (1),n

= mσσ∗ (f )Gp

(T )

where S denotes the σ-algebra generated by (πn )n∈N . Therefore it appears to (1),n be a necessary condition that Gp (T ) converges in order for V (p, f, πn )T to converge as well. This reasoning also carries over to the case of non-constant σs via an approximation of σs by piecewise constant stochastic processes. We then obtain the following result which covers the whole class of positively homogeneous functions f : R2 → R. Theorem 2.17. Let p ≥ 0 and suppose that Condition 1.3 is fulfilled, and that the observation scheme is exogenous and synchronous. Further assume that (1),n

Gp

(2),n

(t) = Gp

P

(t) −→ Gp (t),

t ∈ [0, T ],

for a (possibly random) continuous function Gp : [0, T ] → R≥0 and that we have one of the following two conditions: a) p ∈ [0, 2), b) p ≥ 2 and X is continuous. Then for all continuous positively homogeneous functions f : R2 → R of degree p it holds that P



V (p, f, πn )T −→

T 0

mcs (f )dGp (s)

where cs = σs σs∗ .

T

Remark 2.18. Note that the integral 0 mcs (f )dGp (s) is well-defined as a Lebesgue-Stieltjes integral because mcs (f ) is c` adl` ag as σs is c` adl` ag and because the function t → Gp (t) is by assumption continuous and it is also increasing as the functions t → Gn  p (t), n ∈ N, are all increasing. As a corollary of Theorem 2.17, we directly obtain the following convergence result for the functionals V (l) (p, g, πn )T . Corollary 2.19. Let l = 1 or l = 2, p ≥ 0, and suppose that Condition 1.3 is fulfilled and that the observation scheme is exogenous. Further assume that (l),n

Gp

P

(l)

(t) −→ Gp (t),

t ∈ [0, T ], (l)

for a (possibly random) continuous function Gp : [0, T ] → R≥0 and that we have one of the following two conditions: a) p ∈ [0, 2),

32

2 Laws of Large Numbers

b) p ≥ 2 and X (l) is continuous. Then for all positively homogeneous functions g : R → R of degree p it holds that V

(l)

P

(p, g, πn )T −→ m1 (g)



T 0

(l)

(l)

(σs )p dGp (s).

Remark 2.20. In Chapter 14 of [30] synchronous observation schemes of the form ti,n = ti−1,n + θtni−1,n εi,n are investigated where θn = (θtn )t≥0 is a strictly positive process which is adapted to the filtration (Ftn )t≥0 , where Ftn denotes the smallest σ-algebra containing the filtration (Ft )t≥0 with respect to which X is defined and which has the property that all ti,n are (Ftn )t≥0 -stopping times; compare Definition 14.1.1 in [30]. εi,n is supposed to be an i.i.d. sequence of positive random variables in i for fixed n. The εi,n are independent of the process X and its components. If nθn converges in u.c.p. to some (Ft )t≥0 -adapted process θ and the moments E[(εi,n )p/2 ] = κn p/2 converge to some κp/2 < ∞ then Lemma 14.1.5 in [30] yields P



Gn p (t) −→ κp/2

t 0

(θs )p/2−1 ds =: Gp (t)

∀t ∈ [0, T ],

and using Theorem 14.2.1 in [30] we conclude P

V (p, f, πn )T −→ κp/2



T 0

 mcs (f )(θs )p/2−1 ds =

T 0

mcs (f )dGp (s)

under the assumptions on f and X made in Theorem 2.17. The assumptions on the observation scheme in [30] are weaker than the assumptions made in this paper in the sense that the observation scheme does not have to be exogenous, but there are still strong restrictions on the law of ti,n − ti−1,n . θtni−1,n is known in advance at time ti−1,n and (ti,n − ti−1,n )/θtni,n = εi,n is an exogenous random variable whose law is independent of the other observation times. On the other hand our assumptions allow for observation schemes which do not fulfil the assumptions made in Chapter 14 of [30]. This is due to the fact that we need no analogon to the i.i.d. property of the εi,n . Gn p in general already converges to a linear p/2 p/2 function if the ratio E[n |Ii,n | ]/E[n|Ii,n |] remains constant; compare Lemma 5.3. Asynchronous observation schemes are not considered in [30].  In the case of asynchronous observation times it is more difficult to derive results similar to Theorem 2.17. For functions f which are like f(p1 ,p2 ) , f (p1 ,p2 ) positively

2.2 Normalized Functionals

33

homogeneous with degree p1 in the first argument and with degree p2 in the second argument it holds that



E[V (p1 + p2 , f, πn )T |S] = n(p1 +p2 )/2−1

(1)

(1)

(2)

|Ii,n |p1 /2 |Ij,n |p2 /2

(2)

i,j:ti,n ∨tj,n ≤T (1)

(1)

(2)

(2)

× E[f (|Ii,n |−1/2 Δi,n X (1) , |Ij,n |−1/2 Δj,n X (2) )|S]1{I (1) ∩I (2) =∅} . i,n

j,n

However unlike in the case of synchronous observation times the law of (1)

(1)

(2)

(2)

(|Ii,n |−1/2 Δi,n X (1) , |Ij,n |−1/2 Δj,n X (2) ) is in general not independent of πn as explained in (2.34). The process X toy from (2.31) has the simplest form of all processes for which the functionals discussed in this section yield a non-trivial limit and hence it makes sense to first investigate conditions which grant convergence of V (p1 + p2 , f, πn )T for X toy . Also, the arguments used in the proof of Theorem 2.17 rely on an approximation of σs by piecewise constant processes in time, which then makes it possible to use results for processes like X toy with a constant σs . In particular, we needed that f (Δi,n X toy ) factorizes into a term that depends only on S and a term that is independent of S. This technique of proof can be extended to the asynchronous setting whenever we can find a fixed natural number N ∈ N and functions gk , hk for k = 1, . . . , N such that we can write (1)

(2)

E[f (Δi,n X toy,(1) , Δj,n X toy,(2) )|S] =

N 

(1)

(2)

(2)

(2)

gk (|Ii,n |, |Ij,n |, |Ij,n ∩ Ij,n |)hk (σ (1) , σ (2) , ρ)

(2.40)

k=1

in our toy example, because in that case we have

E[V (p1 + p2 , f, πn )T |S] =

N 

hk (σ (1) , σ (2) , ρ)

k=1

×n

(p1 +p2 )/2−1

 (1)

(2)

i,j:ti,n ∨tj,n ≤T

(1)

(2)

(2)

(2)

gk (|Ii,n |, |Ij,n |, |Ij,n ∩ Ij,n |)1{I (1) ∩I (2) =∅} i,n

j,n

where the right-hand side converges if we assume that the expression in the last line converges as n → ∞ for each k = 1, . . . , N . It is in general not possible to find a representation of the form (2.40) for an arbitrary function f which is positively homogeneous in both arguments. However, there are two interesting cases where such a representation is available. The first

34

2 Laws of Large Numbers

case is when X toy,(1) and X toy,(2) are uncorrelated, i.e. if ρ ≡ 0 on [0, T ], because then (1)

(2)

E[f (Δi,n X toy,(1) , Δj,n X toy,(2) )|S] (1)

(2)

= (σ (1) )p1 (σ (2) )p2 E[f (Z, Z )]|Ii,n |p1 /2 |Ij,n |p2 /2 holds for independent standard normal random variables Z, Z . The result obtained in this case is stated in Theorem 2.21. In the second case we consider the functions f(p1 ,p2 ) with p1 , p2 ∈ N0 . Indeed it holds (1)

(2)

f(p1 ,p2 ) (Δi,n X toy,(1) , Δj,n X toy,(2) ) LS

(1)

(2)

(1)

(2)

= (σ (1) )p1 (σ (2) )p2 (|Ii,n \ Ij,n |1/2 Z1 + |Ii,n ∩ Ij,n |1/2 Z2 )p1 (1)

(2)

× (|Ii,n ∩ Ij,n |1/2 (ρZ2 +



(2)

(1)

1 − ρ2 Z3 ) + |Ij,n \ Ii,n |1/2 Z4 )p2

L

where =S denotes equality of the S-conditional law and Z1 , . . . , Z4 are i.i.d. standard normal random variables. Here the right-hand side can be brought into the form (2.40) using the multinomial theorem. The result obtained in this case is stated in Theorem 2.22. Unfortunately, for the functions f (p1 ,p2 ) there exists no similar representation: however, for even p1 , p2 we have f(p1 ,p2 ) = f (p1 ,p2 ) and Theorem 2.22 also applies there. Theorem 2.21. Let p1 , p2 ≥ 0 and suppose that Condition 1.3 is fulfilled, that the observation scheme is exogenous and that we have ρ ≡ 0 on [0, T ]. Further assume that P

Gn p1 ,p2 (t) −→ Gp1 ,p2 (t), t ∈ [0, T ], for a (possibly random) continuous function Gp1 ,p2 : [0, T ] → R≥0 and that we have one of the following two conditions: a) p1 + p2 ∈ [0, 2), b) p1 + p2 ≥ 2 and X is continuous. Then for all continuous functions f : R2 → R which are positively homogeneous of degree p1 in the first argument and positively homogeneous of degree p2 in the second argument it holds that P

V (p1 + p2 , f, πn )T −→ mI2 (f )



T 0

(1)

(2)

(σs )p1 (σs )p2 dGp1 ,p2 (s).

Here I2 denotes the two-dimensional identity matrix.

(2.41)

2.2 Normalized Functionals

35

Theorem 2.22. Let p1 , p2 ∈ N0 and suppose that Condition 1.3 is fulfilled and that the observation scheme is exogenous. Define L(p1 , p2 ) = {(k, l, m) ∈ (2N0 )3 : k ≤ p1 , l + m ≤ p2 , p1 + p2 − (k + l + m) ∈ 2N0 }.

Assume that for all k, m ∈ N0 for which an l ∈ N0 exists with (k, l, m) ∈ L(p1 , p2 ) there exist (possibly random) continuous functions Hk,m,p1 +p2 : [0, T ] → R≥0 which fulfill P

n Hk,m,p (t) −→ Hk,m,p1 +p2 (t), 1 +p2

t ∈ [0, T ].

Further we assume that we have one of the following two conditions: a) p1 + p2 ∈ {0, 1}, b) p1 + p2 ≥ 2 and X is continuous. Then V (p1 + p2 , f(p1 ,p2 ) , πn )T P



−→

T 0

(1) (2) (σs )p1 (σs )p2

 

 (k,l,m)∈L(p1 ,p2 )

× m1 (x holds. Here



p1 +p2 −(k+l+m)

 p2  l,m

p1 k

 p2 m1 (xk )m1 (xl )m1 (xm ) l, m

)(1 − ρ2s )l/2 (ρs )p2 −(l+m) dHk,m,p1 +p2 (s)

 (2.42)

= p2 !/(l!m!(p2 − l − m)!) denotes the multinomial coefficient.

Example 2.23. In this example we will use Theorem 2.22 to find the limit of V (p1 + p2 , f(p1 ,p2 ) , πn )T for a few non-trivial cases with small p1 , p2 . For p1 = p2 = 1 the set L(1, 1) contains only (0, 0, 0) and we get P

V (2, f(1,1) , πn ) −→ P

as H0,0,2 (t) −→

T 0



T 0



(1) (2)

ρs σs σs dH0,0,2 (s) =

T 0

(1) (2)

ρs σs σs ds

t. Hence V (2, f(1,1) , πn ) converges to the covariation of

X (1) , X (2) for continuous processes X and we have retrieved (2.4) for continuous semimartingales X. For p1 = 1, p2 = 2 we get L(1, 2) = ∅ as k, l, m and P

3 − (k + l + m) cannot be all divisible by 2. Hence V (3, f(1,2) , πn ) −→ 0. This holds for all (p1 , p2 ) where p1 + p2 is odd. Further we define p/2−1 Gn k,m,p (t) = n



(1)

(1)

(2)

|Ii,n |k/2 |Ij,n |m/2

(2)

i,j:ti,n ∨tj,n ≤t (1)

(2)

× |Ii,n ∩ Ij,n |(p−k−m)/2 1{I (1) ∩I (2) =∅} i,n

j,n

36

2 Laws of Large Numbers

and define Gk,m,p (t) as the limit of Gn k,m,p (t) in probability as n → ∞ if it exists. For p1 = p2 = 2 we have to consider the set L(2, 2) = {0, 2} × {(0, 0), (0, 2), (2, 0)} and then obtain using the above notation



P

V (4, f(2,2) , πn ) −→

T 0

(1) (2)

(σs σs )2 (3ρ2s dH0,0,4 (s) + dH0,2,4 (s)

+ (1 − ρ2s )dH0,0,4 (s) + ρ2s dH2,0,4 (s) + dH2,2,4 (s) + (1 − ρ2s )dH2,0,4 (s))



T

= 0

(1) (2)

(σs σs )2 (2ρ2s dH0,0,4 (s) + dG2,2,4 (s))

(2.43)

where we used G2,2,p (s) = H0,0,p (s) + H0,2,p (s) + H2,0,p (s) + H2,2,p (s), p ≥ 4, which follows from the identity (1)

(2)

(1)

(2)

(1)

(2)

(1)

(2)

(2)

(1)

|Ii,n ||Ij,n | = (|Ii,n ∩ Ij,n | + |Ii,n \ Ij,n |)(|Ii,n ∩ Ij,n | + |Ij,n \ Ii,n |). Without presenting detailed computations we state two more results to demonstrate that the limit in Theorem 2.22 after simplification sometimes has a much shorter representation compared to the general form in (2.42). For p1 = p2 = 3 we have L(3, 3) = L(2, 2) and we get after simplification P



V (6, f(3,3) , πn ) −→

T 0

(1) (2)

(σs σs )3 (6ρ3s dH0,0,6 (s) + 9ρs dG2,2,6 (s))

and for p1 = p2 = 4 we obtain V (8, f(4,4) , πn ) P

−→



T 0

(1) (2)

(σs σs )4 (24ρ4s dH0,0,8 (s) + 72ρ2s dG2,2,8 (s) + 9dG4,4,8 (s)).

 Two not very difficult generalizations can be made for the statements in Theorems 2.17, 2.21 and 2.22. The previous results were only stated in a more specific form to keep the notation and the proofs clearer and to direct the reader’s focus to the key aspects. First, throughout this section the rate n was chosen rather arbitrarily as the appropriate scaling factor by which the average interval lengths decrease (l),n and such that we obtain convergence for the functions Gp (t), Gn p1 ,p2 (t) and n Hk,m,p (t). Remark 2.24. Let r : N → [0, ∞) be a function with r(n) → ∞ for n → ∞. Then we obtain the same result as in Corollary 2.19 if we set V

(l)



(p, g, πn )T = (r(n))p/2−1

(l)

g(Δi,n X (l) ),

(l)

(l),n

Gp

(t) = (r(n))p/2−1

 (l)

i:ti,n ≤T

i:ti,n ≤t

(l)

|Ii,n X (l) |p/2 .

2.2 Normalized Functionals

37

Similarly the results from Theorems 2.17, 2.21 and 2.22 hold as well if we replace n by r(n) in the definition of the functional V (p, f, πn )T and the functions Gn p1 ,p2 (t), n Hk,m,p (t). The proofs for these claims are identical to the proofs in the more specific case r(n) = n. Hence we only need that the observation scheme scales with a deterministic rate r(n) to obtain the results in this section.  Further, only increments of the continuous martingale part of X contribute to the limits in Theorems 2.17, 2.21 and 2.22 and the increments of the continuous part of X tend to get very small as the observation intervals become shorter. Hence only function evaluations f (x) at very small x and especially the behaviour of f (x) for x → 0 has an influence on the asymptotics. These arguments motivate that the convergences in the above theorems do not only hold for positively homogeneous functions but as the following corollary shows also for functions f which are very close to being positively homogeneous for x → 0; compare Corollary 3.4.3 in [30]. Corollary 2.25. Suppose that the convergence in one of the Theorems 2.17, 2.21 and 2.22 holds for the function f : R2 → R. Then the corresponding convergence also holds for all functions f˜: R2 → R which can be written as f˜(x) = L(x)f (x) for a locally bounded function L(x) that fulfils limx→0 L(x) = 1. Example 2.26. In the setting of Poisson sampling the assumptions of Theorems (l),n n 2.17, 2.21 and 2.22 are fulfilled. Indeed the functions Gp , Gn p1 ,p2 , Hk,m,p converge by Corollary 5.6 to deterministic linear and hence continuous functions. (l) Although the functions Gp , Gp1 ,p2 , Hk,m,p are in general unknown they can easily (l),n

be estimated by Gp



n , Gn p1 ,p2 , Hk,m,p .

2.2.2 The Proofs We will start this section by deriving some elementary inequalities for positively homogeneous functions. First, we need the following estimate for increments of the functions x → xp . Lemma 2.27. Let p ≥ 0 and x, y ∈ Rd . Then it holds

  x + yp − xp  ≤



yp

p ∈ [0, 1], p

p

p

Kp (ε x + Kp,ε y )

p > 1,

for all ε > 0 where Kp and Kp,ε denote constants depending only on p respectively on p and ε.

38

2 Laws of Large Numbers

Proof. The case p = 0 is trivial. For all a, b ≥ 0 and p ∈ (0, 1] it holds



a

ap = 0



a+b

pz p−1 dz ≥

pz p−1 dz = (a + b)p − bp

b

where the inequality holds because of p > 0 and z → z p−1 being monotonically decreasing. Using this inequality we obtain x + yp ≤ (x + y)p ≤ xp + yp

(2.44)

which yields the claim (together with the analogous inequality for x = (x + y) − y) in the case p ∈ (0, 1]. In the case p > 1 the mean value theorem applied for the function x → xp states that there exists a ξ ∈ [0, 1] with |x + yp − xp | = |(x + y − x)p(ξx + y + (1 − ξ)x)p−1 | ≤ py(x + y + x)p−1 ≤ 2p−1 py(y + x)p−1 where the first inequality follows because x → xp−1 is monotonically increasing on [0, ∞) and hence the maximum of (ξx + y + (1 − ξ)y)p−1 is attained at ξ = 0 or ξ = 1. From Jensen’s inequality we obtain



a+b 2

p−1 ≤

ap−1 bp−1 + 2 2

for p > 2. Using this relation and inequality (2.44) for p ∈ (1, 2] yields |x + yp − xp | ≤ Kp (yp + yxp−1 ). Further we obtain from Muirhead’s inequality, compare Theorem 45 in [19], for a1 = yε−(p−1) , a2 = xε and the exponent vectors (p − 1, 1) ≺ (p, 0) yxp−1 ≤ (yε−(p−1) )(εx)p−1 + (yε−(p−1) )p−1 (εx) ≤ ε−p(p−1) yp + εp xp which yields the claim. Using the above lemma we derive the following estimates for continuous and positively homogeneous functions.

2.2 Normalized Functionals

39

Lemma 2.28. Let f : Rd1 × Rd2 → R, d1 , d2 ∈ N0 , be a continuous function which is positively homogeneous with degree p1 ≥ 0 in the first argument and with degree p2 ≥ 0 in the second argument. Then there exists a constant K with |f (x1 , x2 )| ≤ Kx1 p1 x2 p2 ∀x1 ∈ Rd1 , x2 ∈ Rd2 .

(2.45)

Further there exists a function θ : [0, ∞) → [0, ∞) depending on f and p with θ(ε) → 0 for ε → 0 and a constant Kp1 ,p2 ,ε which may depend on f such that |f (x1 + y1 , x2 + y2 ) − f (x1 , x2 )| ≤ θ(ε)x1 p1 x2 p2 + Kp1 ,p2 ,ε (y1 p1 (x2 p2 + y2 p2 ) + y2 p2 (x1 p1 + y1 p1 ))

(2.46)

holds for all x1 , y1 ∈ Rd1 and x2 , y2 ∈ Rd2 . In the case p2 = 0 the inequality (2.46) can be replaced by |f (x1 + y1 , x2 + y2 ) − f (x1 , x2 )| ≤ θ(ε)x1 p1 + Kp1 ,ε y1 p1

(2.47)

for all x1 , y1 ∈ Rd1 and x2 , y2 ∈ Rd2 . The analogous result holds if p1 = 0. Proof. From the defining property of a positively homogeneous function we obtain f (x1 , x2 ) = x1 p1 x2 p2 f (x1 /x1 , x2 /x2 ) ≤ x1 p1 x2 p2

sup (z1 ,z2 )∈Rd :z1 =z2 =1

|f (z)|.

(2.45) then follows because the function f is continuous which yields that f is bounded on any compact set. Next we prove (2.47). To this end note that for p2 = 0 it holds f (x, y) = lim λ0 f (x, y) = lim f (x, λy) = f (x, 0) λ↓0

(2.48)

λ↓0

by the continuity of f for any x ∈ Rd1 and y ∈ Rd2 . Hence for p2 = 0 the function f does not depend on the second argument at all and for showing (2.47) it suffices to prove that |f˜(x + y) − f˜(x)| ≤ θ(ε)xp1 + Kp1 ,ε yp1

(2.49)

holds for any x, y ∈ R where f˜: R → R denotes a continuous and positively homogeneous function with degree p1 ≥ 0 and θ denotes a function as described above. For proving (2.49) we observe d1

 

d1

 x+y 

 x    x + y x     x  x+y   sup |f˜(z)| + xp1 f˜ − f˜ .

|f˜(x + y) − f˜(x)| = x + yp1 f˜ ≤ |x + yp1 − xp1 |

z∈Rd1 :z=1

− xp1 f˜

x + y

x

(2.50)

40

2 Laws of Large Numbers

Defining ˜ = θ(ε)

sup x,y∈Rd1 :x=y=1∧x−y≤ε

|f˜(x) − f˜(y)|

we obtain   x+y   x+y  x    x  ˜   p1  ˜ + 2 − f˜ sup |f˜(z)| ε−p1  − f   ≤ θ(ε) x + y x x + y x d z∈R 1 :z=1  p1 1 p1 1 yp1 ˜ + Kp ,ε x + yp1  ˜ + Kp ,ε y − ≤ θ(ε) ≤ θ(ε)  + Kp1 ,ε 1 1 x + y x xp1 xp1 where we used (a + b)p1 ≤ Kp1 (ap1 + bp1 ) for all p1 ≥ 0 and a, b ≥ 0 which we showed in the proof of Lemma 2.27. Using this estimate in (2.50) we get p1 ˜ + Kε yp1 |f˜(x + y) − f˜(x)| ≤ K|x + yp1 − xp1 | + θ(ε)x

˜ which together with Lemma 2.27 yields (2.49). Note therefore that θ(ε) → 0 as ε → 0 because f is uniformly continuous on the compact unit sphere. The proof of (2.46) is as the proof of (2.47) mainly based on Lemma 2.27 and uses very similar estimates. It is therefore skipped here. To discuss the synchronous setting and the asynchronous setting simultaneously  of the form (1.1). X  (1) we consider a (d1 + d2 )-dimensional Itˆ o semimartingale X  denotes the vector-valued process containing the first d1 components of X and is (1)  (2) observed at the observation times ti,n , X contains the remaining d2 components (2)

 and is observed at the observation times t . Then the synchronous setting of X i,n and the asynchronous setting discussed in Section 2.2 correspond to d1 = 2, d2 = 0, p1 = p, p2 = 0, and d1 = d2 = 1 respectively. Here the notion of a zero-dimensional semimartingale remains ambiguous. However, we will only plug in increments of this zero-dimensional process into the argument of f in which f is positively homogeneous of degree p2 = 0 and then f constant in this argument; compare (2.48). Hence it is not necessary to specify the notion of a zero-dimensional semimartingale as we are going to use it only to indicate the case where the function f solely depends on the first argument. Further we define 

∗

V (p, f, πn )T = np/2−1

(1)



(1)

(2)



 (1) , Δ X  (2) 1∗,n (i, j) f Δi,n X j,n d1 ,d2

(2)

i,j≥0:ti,n ∨tj,n ≤T

(2.51) for all functions f : Rd1 × Rd2 → R where we set and

1∗,n d1 ,d2 (i, j) =

(l) I0,n

= ∅,

(l) Δ0,n X (l)

= 0, l = 1, 2,

⎧ ⎪ ⎪1{I (1) ∩I (2) =∅} , d1 > 0, d2 > 0, ⎨ i,n

j,n

1{i>0,j=0} , ⎪ ⎪ ⎩1 {i=0,j>0} ,

d1 > 0, d2 = 0, d1 = 0, d2 > 0.

(2.52)

2.2 Normalized Functionals

41

That means, we start the sum in (2.51) at zero and define the indicator 1∗,n d1 ,d2 (i, j) (l)

in such a way that, whenever d3−l = 0, for any i ∈ N with ti,n ≤ T exactly ∗

(l)

 (l) occurs in the sum. Hence V (p, f, πn )T one summand depending on Δi,n X corresponds to V (p, f, πn )T in the asynchronous setting for d1 = d2 = 1 and ∗ V (p, f, πn )T corresponds to V (p, f, πn )T in the synchronous setting for dl = 2, d3−l = 0. Definition 2.29. We denote by ∗

C (p, f, πn )T = np/2−1



(1)



(1)



(2)

(1) , Δ C (2) 1∗,n (i, j) f Δi,n C j,n d1 ,d2

(2)

i,j:ti,n ∨tj,n ≤T ∗

 instead the functional V (p, f, πn )T evaluated at the continuous martingale part C  itself. C(p, f, πn )T is defined as C ∗ (p, f, πn )T above only with C  replaced of at X by C.  As for the indicator in (2.52) we also define a unifying notation for the functions (l),n Gn via p1 ,p2 (t) and Gp (d ,d ),n

p11,p22 G

⎧ n ⎪ ⎨Gp1 ,p2 (t), d1 > 0, d2 > 0,

(t) =

(1),n

⎪ ⎩

Gp1

(t),

d1 > 0, d2 = 0,

Gp2

(t),

d1 = 0, d2 > 0.

(2),n

(2.53)

The following proposition yields that by specifying d1 , d2 appropriately as discussed above, it suffices to prove the convergences in Theorems 2.17, 2.21 and 2.22 for C(p, f, πn )T , C(p1 +p2 , f, πn )T instead of V (p, f, πn )T , V (p1 +p2 , f, πn )T . Proposition 2.30. Let f : Rd1 × Rd2 → R be a function as in Lemma 2.28. p(d11,p,d22 ),n (T ) = OP (1) and let either p1 + p2 ∈ [0, 2) or p1 + p2 ≥ 2 Suppose that G  is continuous. Further we assume dl = 0 ⇒ pl = 0, l = 1, 2. and assume that X Then it holds that ∗

∗

P

V (p1 + p2 , f, πn )T − C (p1 + p2 , f, πn )T −→ 0

(2.54)

as n → ∞.

  M (q), N  (q) a decomposition of X  Proof. In the following, we denote by B(q), C, similar to (1.7). Using (2.46) we obtain

 ∗  V (p1 + p2 , f, πn )T − C ∗ (p1 + p2 , f, πn )T  

(1)  (1) p1 (2)  (2) p2 ≤ n(p1 +p2 )/2−1 1∗,n  Δj,n C  (2.55) d1 ,d2 (i, j) θ(ε)Δi,n C (1)

(2)

i,j:ti,n ∨tj,n ≤T (1)

(2)

(2)

 − C)  (1) p1 (Δ C (2) p2 + Δ (X  − C)  (2) p2 ) + Kp1 ,p2 ,ε Δi,n (X j,n j,n (2)

(1)

(1)



(2.56)

 − C)  (2) p2 (Δ C (1) p1 + Δ (X  − C)  (1) p1 ) . (2.57) + Kp1 ,p2 ,ε Δj,n (X i,n i,n

42

2 Laws of Large Numbers

For (2.55) we get using the Cauchy-Schwarz inequality and inequality (1.9), as Lemma 1.4 holds for Itˆ o semimartingales of arbitrary dimension,



E n(p1 +p2 )/2−1

(1)

(1)

 

(2)

(1) p1 Δ C (2) p2 1∗,n (i, j)S θ(ε)Δi,n C j,n d1 ,d2

(2)

i,j:ti,n ∨tj,n ≤T (d ,d ),n

p11,p22 ≤ θ(ε)K G

(T ). (d ,d ),n

p11,p22 Hence by the assumption G ε→0

n→0

(T ) = OP (1) and Lemma 2.15 we obtain



 lim lim sup P θ(ε)n(p1 +p2 )/2−1

(1)

(1)



(2)

(1) p1 Δ C (2) p2 1∗,n (i, j) > δ = 0 Δi,n C j,n d1 ,d2

(2)

i,j:ti,n ∨tj,n ≤T

for any δ > 0. To prove (2.54) it now remains to show that (2.56) vanishes as n → ∞ for any ε > 0 because then (2.57) can be dealt with analogously by symmetry. We will separately discuss the cases p1 > 0 and p1 = 0. Case 1. We first consider the case where p1 > 0. In the situation p1 + p2 ≥ 2 =B +C  with B t = t ˜bs ds for some bounded process ˜b. Hence we get we have X 0 that the S-conditional expectation of (2.56) is bounded by



Kp1 ,p2 ,ε n(p1 +p2 )/2−1

(1)

(1) (2) i,j:ti,n ∨tj,n ≤T

(d ,d ),n

p11,p22 ≤ Kp1 ,p2 ,ε (|πn |T )p1 /2 G

(2)

(2)

|Ii,n |p1 (|Ij,n |p2 + |Ij,n |p2 /2 )1∗,n d1 ,d2 (i, j)

(T )

which vanishes as n → ∞ for p1 > 0. Next we consider (2.56) in the case p1 + p2 < 2. Using (1)

(1)

(1)

 − C)  (1) p1 ≤ Kp1 (Δ (B(q)  +M (q))(1) p1 + Δ N  (1) (q)p1 ) Δi,n (X i,n i,n (1)

 − C)  separately. Applying allows to treat the different components of Δi,n (X H¨ older’s inequality for p = 2/(2 − p2 ), q = 2/p2 and using the inequalities from Lemma 1.4 (note that p1 < 2 − p2 ) yields 

E n(p1 +p2 )/2−1

(1)

(1)

 +M (q))(1) p1 Kp1 ,p2 ,ε Δi,n (B(q)

(2)

i,j:ti,n ∨tj,n ≤T (2)

 

(2)

(2) p2 + Δ (X  − C)  (2) p2 )1∗,n (i, j)S × (Δj,n C j,n d1 ,d2 ≤ Kp1 ,p2 ,ε n(p1 +p2 )/2−1



(1)

2p1

(1)  (q))(1)  2−p2 |S] E[Δi,n (B(q) +M

(2)

i,j:ti,n ∨tj,n ≤T (2)

(2)

(2) 2 |S]p2 /2 + E[Δ (X  − C)  (2) 2 |S]p2 /2 )1∗,n (i, j) × (E[Δj,n C j,n d1 ,d2 (d ,d ),n

p11,p22 ≤ Kp1 ,p2 ,ε (Kq (|πn |T )p1 /2 + (eq )p1 /2 )G

(T )

2−p2 2

2.2 Normalized Functionals

43

which vanishes as n, q → ∞ for any ε > 0 if p1 > 0. Finally consider



n(p1 +p2 )/2−1 Kp1 ,p2 ,ε

(1)

(1)

 (1) (q)p1 Δi,n N

(2)

i,j:ti,n ∨tj,n ≤T (2)

(2)

(2) p2 + Δ (X  − C)  (2) p2 )1∗,n (i, j) × (Δj,n C j,n d1 ,d2 

≤ n(p1 +p2 )/2−1 Kp1 ,p2 ,ε

(1)

(1)

 (1) (q)p1 Δi,n N

(2)

i,j:ti,n ∨tj,n ≤T (2)

(2)

(2)

 (2) (q)p2 + Δ C (2) p2 + Δ M (2) (q)p2 )1∗,n (i, j) (2.58) × (Δj,n B j,n j,n d1 ,d2 

+ n(p1 +p2 )/2−1 Kp1 ,p2 ,ε

(1)

(2)

 (1) (q)p1 Δ N  (2) (q)p2 Δi,n N j,n

(1) (2) i,j:ti,n ∨tj,n ≤T

× 1∗,n d1 ,d2 (i, j).

(2.59)

(2.59) vanishes as n → ∞ due to p1 + p2 < 2 and because the finitely many jumps of N (q) are asymptotically separated by the observation scheme. Further choose δ > 0 such that p1 ∨ 1 < 2 − δ, 2(p1 + p2 ) + (2 − p2 )δ < 4. Then the S-conditional expectation of (2.58) is by H¨ older’s inequality for p = (2 − δ)/p1



and q = p /(p − 1) and inequalities (1.8)–(1.11) bounded by



n(p1 +p2 )/2−1 Kp1 ,p2 ,ε

(1)





(1)  (1) (q)2−δ |S] E[Δi,n N

p1  2−δ

(2)

1∗,n d1 ,d2 (i, j)

i,j:ti,n ∨tj,n ≤T (2)

 (2) (q) × E[Δj,n B

2−δ p2 2−δ−p 1

(2)

(2)

(2)  + Δj,n C

(2) (q) + Δj,n M

2−δ p2 2−δ−p



≤ n(p1 +p2 )/2−1 Kp1 ,p2 ,ε

(1)

1

|S]





2−δ p2 2−δ−p

1

2−δ−p1 2−δ

(1)

p1

(1)



(2)

Kq |Ii,n | 2−δ + |Ii,n |p1 Kq |Ij,n |

p2 2

(2)

i,j:ti,n ∨tj,n ≤T

× 1∗,n d1 ,d2 (i, j) p1

≤ Kp1 ,p2 ,ε,q (|πn |T ) 2−δ −

p1 2

(d ,d ),n

p11,p22 G

(T )

where we used p2 q = p2

p

2−δ < 2 ⇔ 2(p1 + p2 ) + (2 − p2 )δ < 4. = p2 p − 1 2 − δ − p1

Hence (2.58) and then also (2.56) vanish by Lemma 2.15 as n → ∞ for any ε and any q > 0 if p1 > 0.

44

2 Laws of Large Numbers

Case 2. Now we consider the case p1 = 0. As (2.54) is trivial for p1 = p2 = 0 it remains to discuss p1 = 0, p2 > 0. In that case ∗

∗

|V (p1 + p2 , f, πn )T − C (p1 + p2 , f, πn )T | is by (2.47) (or rather the analogous result with p1 = 0 and p2 > 0) bounded by n(p1 +p2 )/2−1

 (1)

(2)

(2)



(2)

(2) p2 + Kp1 ,p2 ,ε Δ (X  − C)  (2) p2 1∗,n (i, j). θ(ε)Δj,n C j,n d1 ,d2

i,j:ti,n ∨tj,n ≤T

Here the first term in the sum corresponds to (2.55) and the second term to (2.57). Hence the term (2.56) does not have to be dealt with if p1 = 0 because such a term simply does not occur in the upper bound in this situation. By symmetry (2.57) can be discussed in the same way as (2.56) with the difference that for (2.57) we have to discuss the cases p2 > 0 and p2 = 0 separately. Hence (2.54) follows because we have shown that (2.55)–(2.57) vanish. Next, we define discretizations of σ and C by σ(r)s = σ(k−1)T /2r ,



C(r)t =

s ∈ [(k − 1)T /2r , kT /2r ),

t 0

(2.60) σs (r)ds.

 for the (d1 + d2 )-dimensional process Similarly we define discretizations of σ ˜ and C  and denote X C

∗,r

(p, f, πn )T = np/2−1

 (1)

(2)



(1)

(2)



(1) (r), Δ C (2) (r) 1∗,n (i, j). f Δi,n C j,n d1 ,d2

i,j:ti,n ∨tj,n ≤T

Proposition 2.31. Let f : Rd1 × Rd2 → R be a function as in Lemma 2.28 and p(d11,p,d22 ),n (T ) = OP (1). Further we assume dl = 0 ⇒ pl = 0, l = 1, 2. assume that G Then ∗

lim lim sup P(|C (p1 + p2 , f, πn )T − C

r→∞ n→∞

holds for any δ > 0.

∗,r

(p1 + p2 , f, πn )T | > δ) = 0

2.2 Normalized Functionals

45

Proof. We obtain using inequality (2.46)

 

∗ ∗,r E |C (p1 + p2 , f, πn )T − C (p1 + p2 , f, πn )T |S 

(1)  (1) (2)  (2) ≤ n(p1 +p2 )/2−1 E |f (Δ C ,Δ C ) i,n

(1)

j,n

(2)

i,j:ti,n ∨tj,n ≤T (1)



≤ n(p1 +p2 )/2−1

(1)

 

(2)

(1) (r), Δ C (2) (r))|S 1∗,n (i, j) − f (Δi,n C j,n d1 ,d2

 (1)  (1) p1 (2)  (2) p2   1∗,n  |Δj,n C  S d1 ,d2 (i, j) E θ(ε)Δi,n C

(2)

i,j:ti,n ∨tj,n ≤T (1)

(2)

(2)

 

(2)

(1)

(1)

 

(1) p1  − C(r))  (2) p2 + Δ C (2) (r)p2 )S + Kε E Δi,n (C  (Δj,n C j,n

(2) p2  − C(r))  (1) p1 + Δ C (1) (r)p1 )S + Kε E Δj,n (C  (Δi,n C i,n (d ,d ),n

p11,p22 ≤ θ(ε)G



(T ) + Kε n(p1 +p2 )/2−1



l=1,2 i,j:t(l) ∨t(3−l) ≤T i,n

E





(l) ti,n (l)

˜ σs − σ ˜s (r)2 ds

ti−1,n

j,n

p pl ∨ 12   12 ∧pl (3−l) 3−l S K|Ij,n | 2 1∗,n d1 ,d2 (i, j)

(2.61)

where we applied the Cauchy-Schwarz inequality, inequality (1.8) and (2.1.34) from [30] together with Jensen’s inequality for pl < 1/2. Using the trivial inequality ax ≤ η˜x + η˜x−1 a which holds for any η˜, a > 0 and x ∈ [0, 1] for x = (1/2) ∧ pl ≤ 1, η˜ =

(l) (η|Ii,n |)pl /(1∧2pl ) ,

a=E

yields that (2.61) is bounded by (note (d ,d ),n

p11,p22 (θ(ε) + Kε η pl /2 )G



(l)

ti,n

˜ σs − σ ˜s (r)2 ds

(l) ti−1,n

((1/2)∧pl −1)pl 1∧2pl

pl ∨ 12   S

+ (pl ∨ 12 ) =

pl 2)

(T )



+ Kε,η n(p1 +p2 )/2−1





(l)

|Ii,n |

((1/2)∧pl −1)pl 1∧2pl

l=1,2 i,j:t(l) ∨t(3−l) ≤T i,n

×E





(l) ti,n (l) ti−1,n

˜ σs − σ ˜s (r)2 ds

j,n

pl ∨ 12   (3−l) p3−l ∗,n S |I 2 1d1 ,d2 (i, j) j,n | (d ,d ),n

p11,p22 ≤ (θ(ε) + Kε η pl /2 + Kε,η (δ 2p1 ∨1 + δ 2p2 ∨1 ))G ×



l=1,2

E





(l) (3−l) i,j:ti,n ∨tj,n ≤T

(l)

|Ii,n |

((1/2)∧pl −1)pl 1∧2pl

(3−l)

|Ij,n

|

(T ) + Kε,η n(p1 +p2 )/2−1

p3−l 2

1∗,n d1 ,d2 (i, j)

46

2 Laws of Large Numbers

×





(l)

ti,n (l)

˜ σs − σ ˜s (r)2 ds

pl ∨ 12

ti−1,n

1{sup

  

s∈(t

(l) (l) ,t ] i−1,n i,n

˜ σs −˜ σs (r)>δ} S

. (2.62)

Denote by Ω(N, n, r, δ) the set where σ ˜ has at most N jump times Sp in [0, T ] with Δ˜ σSp  > δ/2, two different such jumps are further apart than |πn |T and ˜ σt − σ ˜s  ≤ δ for all s, t ∈ [0, T ] with s < t, |t−s| < 2−r +|πn |T and p : Sp ∈ [t, s]. Then P(Ω(N, n, r, δ)) → 1 as N, n, r → ∞ for any δ > 0 because σ is c` adl` ag. Using the assumption that σ ˜ is bounded we get that (2.62) is less or equal than





p(d11,p,d22 ),n (T ) θ(ε) + Kε η pl /2 + Kε,η [δ 2p1 ∨1 + δ 2p2 ∨1 + P((Ω(N, n, r, δ))c |S)] G  (d1 ,d2 ),n  p(d11,p,d22 ),n (s) p1 ,p2 +Kε,η N sup G (t) − G 0≤s ε ) = 0

for all ε > 0. Proof of Theorem 2.17. By Proposition 2.30 it suffices to prove Theorem 2.17 only in the case Xt = Ct . We consider the discretization (2.60) and denote cs (r) = σs (r)σs (r)∗ . Setting Rn = C(p, f, πn )T ,



T

R=

mcs (f )dGp (s),

0

Rn (r) = np/2−1







f Δi,n C(r) ,

i:ti,n ≤T



T

R(r) = 0

mcs (r) (f )dGp (s),

we will prove















lim lim sup P R − R(r) + R(r) − Rn (r) + Rn (r) − Rn  > δ = 0 ∀δ > 0.

r→∞ n→∞

Step 1. As cs is c` adl` ag and Gp (s) is continuous it holds



T

R= 0

mcs− (f )dGp (s).

2.2 Normalized Functionals

47

Deonte by Φ0,I2 the distribution function of a two-dimensional standard normal random variable. Further consider a function ψ : [0, ∞) → [0, 1] as in the proof of Theorem 2.3 with 1[1,∞) (x) ≤ ψ(x) ≤ 1[1/2,∞) (x) and define ψA (x) = ψ(x/A)

and ψA = 1 − ψA for A > 0. Note that

  |R − R(r)| = 

0 T

 ≤

T





 

R2

R2

0



T

(f (σs− x) − f (σs (r)x))Φ0,I2 (dx)dGp (s)

|(f ψA )(σs− x) − (f ψA )(σs (r)x)|Φ0,I2 (dx)dGp (s)



+ 0

R2



|(f ψA )(σs− x) − (f ψA )(σs (r)x)|Φ0,I2 (dx)dGp (s). (2.63)



By (2.45) for p1 = p, p2 = 0 we obtain |(f ψA )(x)| ≤ KAp and hence (f ψA ) is

bounded. Then the fact that (f ψA ) is continuous together with the pointwise convergence σs (r) → σs− yields by dominated convergence that the second summand in (2.63) vanishes for any A > 0 as r → ∞. The first summand in (2.63) is bounded by



T





K R2

0



σs− xp 1{σs− x≥A/2} + σs (r)xp 1{σs (r)x≥A/2} Φ0,I2 (dx)dGp (s)

(2.64) where the inner integral is increasing in σs . As we assume that σ is bounded on [0, T ] this yields that there exists a constant K =

(1)

(2)

ess sup (|σs (ω)| + |σs (ω)|)

s∈[0,T ],ω∈Ω

such that (2.64) is bounded by





K R2



K xp 1{K  x≥A/2} + K xp 1{K  x≥A/2} Φ0,I2 (dx)



T 0

dGp (s)

which vanishes as A → ∞. Hence we have shown |R − R(r)| → 0 almost surely as r → ∞. P Step 2. In order to prove |R(r) − Rn (r)| −→ 0 as n → ∞ we apply Lemma C.2 with ξkn = np/2−1

 i∈L(n,k,T )

f (Δi,n C(r)),

48

2 Laws of Large Numbers

rn rn rn L(n, k, T  ) = {i : ti−1,n ∈ [(k − 1)T /2 , kT /2 )}, k = 1, 2, . . . , 2 , and set n Gk = σ F(k−1)T /2rn ∪ S . Here, rn is a sequence of real numbers with rn ≥ r, rn → ∞ and (l),n

2rn sup |Gp (s) − Gp s∈[0,T ]

2 rn

(s)| = oP (1), (l),n

sup s,t∈[0,T ],|t−s|≤|πn |T

|Gp

(l),n

(t) − Gp

(2.65) (s)| = oP (1).

(l),n

Such a sequence exists, because Gp and hence Gp are nondecreasing functions such that pointwise convergence implies uniform convergence on [0, T ] to the continuous function Gp . We then get



 n  E ξkn Gk−1 = np/2−1 =n

p/2−1

=n

p/2−1

 n 

E f (Δi,n C(r))Gk−1

i∈L(n,k,T )





n |Ii,n |p/2 E f (|Ii,n |−1/2 Δi,n C(r))Gk−1

i∈L(n,k,T )



mc(k−1)T /2rn (r) (f ) i:Ii,n

+K



+K

|Gp

(l),n

= mc(k−1)T /2rn (r) (f ) Gp

(l),n

(t) − Gp

(s)|

(l),n

(kT /2rn ) − Gp (l),n

sup s,t∈[0,T ],|t−s|≤|πn |T

|Ii,n |p/2

⊂((k−1)T /2rn ,kT /2rn ]

(l),n

sup s,t∈[0,T ],|t−s|≤|πn |T

|Gp



(l),n

(t) − Gp

((k − 1)T /2rn )



(s)|

n because |Ii,n |−1/2 Δi,n C(r) is conditional on Gk−1 centered normal distributed with covariance matrix c(k−1)T /2rn (r). The term

K

(l),n

sup s,t∈[0,T ],|t−s|≤|πn |T

|Gp

(l),n

(t) − Gp

(s)|

is due to the summand with kT /2rn ∈ Ii,n which has to be treated separately as in the corresponding interval the process σ(r) might jump. Further as in Step 1 the boundedness of σs together with |f (x)| ≤ Kxp yields that mcs (f ) is also bounded which together with the previous computations yields 2    n R(r) − E[ξkn |Gk−1 ] rn

k=1 rn ≤ K2rn sup |Gp (s)−Gn p (s)|+K2 s∈[0,T ]

sup s,t∈[0,T ],|t−s|≤|πn |T

(l),n

|Gp

(l),n

(t)−Gp

(s)|

2.2 Normalized Functionals

49



n where the right-hand side is oP (1) by (2.65). Hence the sum over the E[ξkn Gk−1 ] converges in probability to R(r). Using the Cauchy-Schwarz inequality, inequality (1.9), the definition of Gn p and telescoping sums we also get 2  rn

2 

 ≤ E np/2−1 rn

n E[|ξkn |2 |Gk−1 ]

k=1

=K

≤K

≤K

≤K

 n 

E Δi,n C(r)p Δj,n C(r)p Gk−1  



i∈L(n,k,T ) j∈L(n,k,T )



np−2



 n 1/2

E Δm,n C(r)2p Gk−1

m=i,j

|Ii,n |p/2 |Ij,n |p/2

i∈L(n,k,T ) j∈L(n,k,T )

k=1 2 rn 





np−2

k=1 2 rn 

2  n  Gk−1

i∈L(n,k,T ) j∈L(n,k,T )

k=1 2 rn 



np−2

KΔi,n C(r)p

i∈L(n,k,T )

k=1 2 rn 





np/2−1



|Ii,n |p/2

2

i∈L(n,k,T )

k=1

≤ KGn p (T )

 (l),n  Gp (u) − G(l),n (s) p

sup u,s∈[0,T ],|u−s|≤T 2−rn +|π

n |T

where the right-hand side converges to zero in probability, since Gn p converges uniformly to a continuous function Gp . Hence we have shown 2  rn

2  rn

P

n E[ξkn |Gk−1 ] −→ R(r),

k=1

P

n E[|ξkn |2 |Gk−1 ] −→ 0.

k=1

Further the ξkn are Gkn -measurable and hence ξkn − E[ξkn |Gn k−1 ] are martingale differences. Lemma C.2 then yields 2  rn

Rn (r) =

P

ξkn −→ R(r)

k=1

for any r ∈ N. Step 3. Finally we obtain lim lim sup P(|Rn − Rn (r)| > δ) = 0

r→∞ n→∞

for any δ > 0 from Proposition 2.31 with d1 = 2, d2 = 0, p1 = p and p2 = 0.

50

2 Laws of Large Numbers

Proof of Corollary 2.19. This is Theorem 2.17 with the function f (x1 , x2 ) = sgn(g(x1 ))|g(x1 )g(x2 )|1/2 (l)

(l)

t = (X , X )∗ . Here, sgn(x) denotes the signum applied to the process X t t function of a real number x ∈ R. Further note that any positively homogeneous function g in dimension 1 is continuous because it holds g(x) = |x|g(1)1{x>0} + |x|g(−1)1{x δ) = 0

r→∞ n→∞

for any δ > 0 from Proposition 2.31 with d1 = d2 = 1. Proof of Theorem 2.22. Proposition 2.30 yields that it suffices to prove Theorem 2.22 in the case Xt = Ct . Further, following the proof of Theorem 2.21 we observe that Step 1 and Step 3 do not make use of the assumption ρ = 0. Hence the arguments therein also apply here. The only difference occurs if we want to adapt Step 2 in the proof of Theorem 2.21 for the proof of Theorem 2.22. In fact, if we look at



ξkn = n(p1 +p2 )/2−1

(i,j)∈L(n,k,T )

(1)

(2)

(Δi,n C (1) (r))p1 (Δj,n C (2) (r))p2 1{I (1) ∩I (2) =∅} i,n

j,n

and denote by Φ0,I4 the distribution function of a four-dimensional standard normal random variable we get  (1) (2) n E[ξkn |Gk−1 ] = n(p1 +p2 )/2−1 E[(Δi,n C (1) (r))p1 (Δj,n C (2) (r))p2 |Gkn ] (i,j)∈L(n,k,T )

=



(l) (σ(k−1)T /2rn )pl

l=1,2



(1)



 (i,j)∈L(n,k,T )

 R4

× 1{I (1) ∩I (2) =∅} i,n

(1) |Ii,n

\

j,n

(2) Ij,n |1/2 x1

(1)

(2)

+ |Ii,n ∩ Ij,n |1/2 x2

(2)

(1)

 p1

(2)

× ρ(k−1)T /2rn |Ii,n ∩ Ij,n |1/2 x2 + (1 − (ρ(k−1)T /2rn )2 )1/2 |Ii,n ∩ Ij,n |1/2 x3  p2 (2) (1) + |Ij,n \ Ii,n |1/2 x4 Φ0,I4 (dx)1{I (1) ∩I (2) =∅} i,n

+K

sup s,t∈[0,T ],|t−s|≤3|πn |T

=

(1) (2) (σ(k−1)T /2rn )p1 (σ(k−1)T /2rn )p2



×

(1)

R4

(2)

j,n

n |Gn p1 ,p2 (t) − Gp1 ,p2 (s)|



p1  p2 

(i,j)∈L(n,k,T ) k=0 l,m=0 (1)



p1 k



p2 l, m



(2)

|Ii,n \ Ij,n |k/2 (x1 )k (ρ(k−1)T /2rn )p2 −(l+m) |Ii,n ∩ Ij,n |(p1 +p2 −(k+l+m))/2 (1)

(2)

× (x2 )p1 +p2 −(k+l+m) (1 − (ρ(k−1)T /2rn )2 )l/2 |Ii,n ∩ Ij,n |l/2 (x3 )l

2.2 Normalized Functionals (2)

53

(1)

× |Ij,n \ Ii,n |m/2 (x4 )m Φ0,I4 (dx)1{I (1) ∩I (2) =∅} i,n

+K

sup s,t∈[0,T ],|t−s|≤3|πn |T

=

j,n

n |Gn p1 ,p2 (t) − Gp1 ,p2 (s)|

(1) (2) (σ(k−1)T /2rn )p1 (σ(k−1)T /2rn )p2



 (k,l,m)∈L(p1 ,p2 )

p1 k



p2 m1 (xk )m1 (xl ) l, m

p1 +p2 −(k+l+m)

× m1 (x )m1 (x )(ρ(k−1)T /2rn )p2 −(l+m) (1 − ρ2(k−1)T /2rn )l/2   n n × Hk,m,p (kT /2rn ) − Hk,m,p ((k − 1)T /2rn 1 +p2 1 +p2 m

+K

sup s,t∈[0,T ],|t−s|≤3|πn |T

n |Gn p1 ,p2 (t) − Gp1 ,p2 (s)|

where we used the multinomial theorem, E[X k ] = 0 for X ∼ N (0, 1), k odd, and



(1)

Δi,n C (1) (r)



LG n

k−1

=

(2) Δj,n C (2) (r)

with

⎛ ⎜

(1)

(1)

(2)

⎞∗

(1)

(1)

(2)

⎟

σ(k−1)T /2rn |Ii,n \ Ij,n |1/2

  v1 v2

U

1/2 ⎟ ⎜ v1 = ⎜σ(k−1)T /2rn |Ii,n ∩ Ij,n | ⎟ ⎝ ⎠ 0 0

⎛

⎞∗

0

(2) (1) (2) ⎜ ⎟ ρ(k−1)T /2rn σ(k−1)T /2rn |Ii,n ∩ Ij,n |1/2 ⎜ ⎟ ⎜ v2 = ⎜ (1) (2) 1/2 ⎟ 2 1/2 (2) ⎟ (1 − (ρ ) ) σ |I ∩ I | r n (k−1)T /2 j,n (k−1)T /2rn i,n ⎝ ⎠ (2)

(2)

(1)

(σ(k−1)T /2rn )|Ij,n \ Ii,n |1/2

(1)

(2)

for Ii,n ∪ Ij,n ⊂ [(k − 1)T /2rn , kT /2rn ) where U = (U1 , U2 , U3 , U4 )∗ is N (0, I4 )n n distributed and independent of F . Here, LGk−1 denotes identity of the Gk−1 conditional distributions. The rest of Step 2 from the proof of Theorem 2.21 also applies here without modification. Remark 2.32. Propositions 2.30 and 2.31 suggest that similar results as in Theorems 2.17, 2.21 and 2.22 could also be derived for (d1 + d2 )-dimensional Itˆ o semimartingales X = (X (1) , X (2) ) where all components of the dl -dimensional process X (l) , l = 1, 2, are observed snychronously and functions f : Rd1 +d2 → R. However, as interesting properties and challenges due to the asynchronicity of the observation scheme already arise in the bivariate setting, I decided to restrict myself to the bivariate setting also to keep the notation and statements clearer. Further, in Part II we are going to discuss applications only for bivariate Itˆ o semimartingales as well. 

54

2 Laws of Large Numbers

Proof of Corollary 2.25. It suffices to prove ∗ ∗ P V (p1 + p2 , f, πn )T − V (p1 + p2 , f˜, πn )T −→ 0.

For ε > 0 pick δ > 0 such that |L(x) − 1| < ε for all x with x ∈ [0, δ]. Then it holds

 ∗  V (p1 + p2 , f, πn )T − V ∗ (p1 + p2 , f˜, πn )T    (1)  (1) (2)  (2) = n(p1 +p2 )/2−1 (1 − L(Δ X ,Δ X )) i,n

(1)

j,n

(2)

i,j:ti,n ∨tj,n ≤T (1)

(2)



 (1) , Δ X  (2) )1∗,n (i, j) × f (Δi,n X j,n d1 ,d2

∗

≤ εV (p1 + p2 , |f |, πn )T



+ n(p1 +p2 )/2−1

(1)

(1)

(2)

 (1) , Δ X  (2) )| |1 − L(Δi,n X j,n

(2)

i,j:ti,n ∨tj,n ≤T (1)

(2)

 (1) , Δ X  (2) )|1 1∗,n (i, j). (2.66) × |f (Δi,n X (1)  (1) (2)  (2) j,n {(Δ X ,Δ X )>δ} d1 ,d2 i,n

j,n

The function |f | : x → |f (x)| is positively homogeneous of degree p1 in the first argument and positively homogeneous of degree p2 in the second argument. Hence using Proposition 2.30 and ∗

E[C (p1 + p2 , |f |, πn )T |S]  (1)  (1) p1 (2)  (2) p2 ≤ K E[|Δi,n C | |Δj,n C | |S]1∗,n d1 ,d2 (i, j) (1)

(2)

i,j:ti,n ∨tj,n ≤T (d ,d ),n

p11,p22 ≤ KG

(T ), ∗

which follows by (2.45), we obtain V (p1 + p2 , |f |, πn )T = OP (1) as n → ∞ and  is continuous, hence the first term in (2.66) vanishes as n → ∞ and then ε → 0. If X  may then the second term in (2.66) converges almost surely to 0 as n → ∞. If X be discontinuous we have p1 + p2 < 2. We then denote by Ω(n, q, N, δ) the set (1)  (1) (2)  (2) (1) (2) where (Δi,n X , Δj,n X ) > δ implies Δi,n N (1) (q) = 0 or Δj,n N (2) (q) = 0, (1)

(2)

s  ≤ N for all s ∈ [0, T ] and (Δ X  (1) , Δ X  (2) ) ≤ 2N for all it holds ΔX i,n j,n (1)

(2)

i, j with ti,n ∨ tj,n ≤ T . On this set the second term in (2.66) is by the local boundedness of L bounded by n(p1 +p2 )/2−1

 (1)

(1)

(2)

 (1) , Δ X  (2) )| K|f (Δi,n X j,n

(2)

i,j:ti,n ∨tj,n ≤T

× 1{(Δ(1) N (1) (q),Δ(2) N (2) (q)) =∅} 1∗,n d1 ,d2 (i, j) i,n

j,n

2.3 Functionals of Truncated Increments

55

which vanishes due to p1 + p2 < 2 using (2.45) and the arguments used for the discussion of (2.58) and (2.59) in the proof of Proposition 2.30. The proof is then finished by observing P(Ω(n, q, N, δ)) → 1 as n, q, N → ∞ for any δ > 0.

2.3 Functionals of Truncated Increments In Section 2.1 only the jump part of X contributes to the limit while in Section 2.2 only the continuous part of X contributes to the limit. Hence for the non(l) normalized functionals only large increments Δi,n X (l) contribute in the limit while (l)

for the normalized functionals only small increments Δi,n X (l) contribute in the limit. Following up on these observations one might expect that expressions of the form   (1)  (2) f Δi,n X (1) , Δj,n X (2) 1{I (1) ∩I (2) =∅} 1{|Δ(1) X (1) |,|Δ(2) X (2) | large“} , (1)

i,n

(2)

i,j:ti,n ∨tj,n ≤T

np/2−1



(1)

(2)



(1)

(2)

j,n

i,n

j,n

”



f Δi,n X (1) , Δj,n X (2) 1{I (1) ∩I (2) =∅} 1{|Δ(1) X (1) |,|Δ(2) X (2) | i,n

i,j:ti,n ∨tj,n ≤T

j,n

i,n

j,n

small“} ”

(2.67) converge to the same limits as V (f, πn )T and V (p, f, πn )T and that the convergence of sums based on selected summands might be faster than for the sums which include all summands. Furthermore we may hope that we obtain convergence for functionals of the form (2.67) for a wider class of functions f than we did in Sections 2.1 and 2.2 for the functionals V (f, πn )T and V (p, f, πn )T . This is due to the fact that most of the conditions on the functions f made in Sections 2.1 and 2.2 were needed to show that the contribution of the continuous respectively of the jump part vanishes in the limit. But these contributions mainly stem from small“ ” respectively large“ increments which we here explicitly exclude from the sums. ” It remains to specify which increments we classify to be small and which to be large. We therefore recall the discussions from the beginning of Section 2.2 (l) were we observed that the increments of the jump part over an interval Ii,n are constant in magnitude as |πn |T → 0 while the increments of the continuous part (l) are of magnitude |Ii,n |1/2 as |πn |T → 0. Hence by checking whether (l)

(l)

|Δi,n X| ≤ β|Ii,n | for some β > 0 and  ∈ (0, 1/2) is fulfilled or not we are asymptotically able to (l) distinguish whether Δi,n X (l) is dominated by the jump part or the continuous

part of X (l) ; compare also Chapter 9 of [30]. Here the rate  ∈ (0, 1/2) should lie between the rate 0 by which increments of the jump parts decrease (or rather

56

2 Laws of Large Numbers

remain constant) and the rate 1/2 by which increments of the continuous part decrease. Building on this motivation we define the functionals



V+ (f, πn , (β, ))T =

(1)



(1)

(2)



f Δi,n X (1) , Δj,n X (2) 1{I (1) ∩I (2) =∅}

(2)

i,n

i,j:ti,n ∨tj,n ≤T

j,n

× 1{|Δ(1) X (1) |>β|I (1) | ,|Δ(2) X (2) |>β|I (2) | } , i,n

V − (p, f, πn , (β, ))T = n



p/2−1

(1)

f



i,n

j,n

j,n

 (1) (2) Δi,n X (1) , Δj,n X (2)

(2)

i,j:ti,n ∨tj,n ≤T

× 1{I (1) ∩I (2) =∅} 1{|Δ(1) X (1) |≤β|I (1) | ,|Δ(2) X (2) |≤β|I (2) | } i,n

j,n

i,n

i,n

j,n

j,n

for functions f : R2 → R and (l)

V+ (g, πn , (β, ))T =

 (l)



(l)



g Δi,n X (l) 1{|Δ(l) X (l) |>β|I (l) | } , i,n

i:ti,n ≤T



(l)

V − (p, g, πn , (β, ))T = np/2−1

(l)



(l)

i,n



g Δi,n X (l) 1{|Δ(l) X (l) |≥β|I (l) | } , i,n

i:ti,n ≤T

i,n

for functions g : R → R and l = 1, 2.

2.3.1 The Results As discussed before Remark 2.7 we cannot get convergence of the non-normalized functionals for functions f that do not fulfil f (x, y) = O((x, y)2 ) as (x, y) → 0 because for such f the quantities B(f ), B ∗ (f ) are in general not well-defined. Hence in Theorem 2.1 and Corollary 2.2 we only get an improvement from functions f with f (x, y) = o(x2 ) as x → 0 to functions with f (x) = O(x2 ) as x → 0. Theorem 9.1.1 in [30] states this result only for equidistant observation times (1) (2) ti,n = ti,n = i/n. In the following theorem we extend their result to the setting of irregular, exogenous and synchronous observations. Theorem 2.33. Suppose the observation scheme is exogenous, synchronous and Condition 1.3 holds. Further let β > 0 and  ∈ (0, 1/2). Then we have with the notation from (2.1)



P

f (Δi,n X)1{Δi,n X>β|Ii,n | } −→ B(f )T

i:ti,n ≤T

for all continuous functions f : R2 → R with f (x) = O(x2 ) as x → 0. (l)

Again we obtain the convergence result for V+ (g, πn , (β, ))T as a corollary.

2.3 Functionals of Truncated Increments

57

Corollary 2.34. Let β > 0,  ∈ (0, 1/2) and assume that Condition 1.3 is fulfilled and that the observation scheme is exogenous. Then it holds P

(l)

V+ (g, πn , (β, ))T −→ B (l) (g)T for all continuous functions g : R → R with g(x) = O(x2 ) as x → 0. In the asynchronous setting in Theorems 2.3 and 2.8 the order condition on f is way stronger compared to the theoretical bound of f (x, y) = O((x, y)2 ) as (x, y) → 0 than in the synchronous setting. Hence we might expect a larger improvement for the condition on f in the asynchronous setting than in the synchronous setting when using the functionals V+ (f, πn , (β, ))T instead of V (f, πn , )T . However in the setting of Theorem 2.3 we will get no improvement at all which will be illustrated in Example 2.36. In Theorem 2.8 on the other hand we can relax the assumption (2.8). In the general case we get an improvement if p1 ∨ p2 < 2, while we always get an improvement in the case where X has only finite jump activity. The improvement is larger if  is smaller i.e. if increments are classified to be large via a higher threshold. The relaxation of (2.8) leads to the fact that the class of functions f for which we obtain convergence under an identical condition is larger for V+ (f, πn , (β, ))T than for V (f, πn )T ; compare Remark 2.37. Theorem 2.35. Suppose Condition 1.3 is fulfilled and let β > 0 and  ∈ (0, 1/2). a) It holds P

V+ (f, πn , (β, ))T −→ B ∗ (f )T for all continuous functions f : R2 → R with f (x, y) = O(x2 y 2 ) as |xy| → 0. b) Let p1 , p2 > 0 with p1 + p2 ≥ 2. Further assume that the observation scheme is exogenous and that we have one of the following: (i) We have

 (1)

(1)

|Ii,n |

p1 +(1/2−)(2−p1 ∨p2 )+ 2

∧1

(2)

i,j:ti,n ∨tj,n ≤T (2)

× |Ij,n |

p1 +(1/2−)(2−p1 ∨p2 )+ 2

∧1

1{I (1) ∩I (2) =∅} = OP (1) i,n

(2.68)

j,n

as n → ∞ where (2 − p1 ∨ p2 )+ = max{2 − p1 ∨ p2 , 0} denotes the positive part of 2 − p1 ∨ p2 .

58

2 Laws of Large Numbers (ii) We have

 (1)

(1)

|Ii,n |

p1 +(1/2−) 2

(2)

|Ij,n |

p2 +(1/2−) 2

1{I (1) ∩I (2) =∅} = OP (1) i,n

(2)

i,j:ti,n ∨tj,n ≤T

j,n

(2.69) and X has almost surely only finitely many jumps in [0, T ]. Then it holds P

V+ (f, πn , (β, ))T −→ B ∗ (f )T for all continuous functions f : R2 → R with f (x, y) = o(|x|p1 |y|p2 ) as |xy| → 0. The following example shows that if V+ (f, πn , (β, ))T is supposed to converge for any Itˆ o semimartingale X and any observation scheme fulfilling Condition 1.3 then the function f has to fulfil the same order condition f (x, y) = O(x2 y 2 ) for |xy| → 0 as was required in Theorem 2.3. Example 2.36. Let X (1) = 1{t≥U } , U ∼ U[0, 1], and the observation scheme (1)

(2)

be given by ti,n = i/n and ti,n = i/n1+γ , γ > 0, as in Example 2.5 and

f (x, y) = |x|p1 |y|p2 , p1 , p2 ≥ 0. Further let X (2) be an α-stable L´evy motion, α ∈ (0, 2), as defined in Example 3.1.3 of [43] i.e. X (2) is a stationary process with independent increments and (2)

Xt

(2) L

− Xs

(2)

= (t − s)1/α X1

∀t ≥ s ≥ 0. (2)

Then for n large enough and i.i.d. random variables Zi ∼ X1 n1+γ nU /n/n1+γ



V+ (f, πn , (β, ))1 =

it holds

 (2) p 2 X 1+γ − X (2)  i/n (i−1)/n1+γ

i=n1+γ (nU −1)/n/n1+γ +1

× 1{|Δ(2) X (2) |>βn−(1+γ) } i,n

n1+γ (nU −1)/n+nγ 



≥

i=n1+γ (nU −1)/n+1 L

p2 −(1+γ)p2

=β n

β p2 n−(1+γ)p2 1{|Δ(2) X (2) |>βn−(1+γ) } i,n

nγ 

 i=1

1{|Zi |>β(n(1+γ)(1/α−) } .

2.3 Functionals of Truncated Increments

59

Using the estimate (1.2.8) from [43] then yields that the expectation of the above quantity is as n → ∞ approximately equal to β p2 n−(1+γ)p2 nγ K(β(n(1+γ)(1/α−) )−α for some constant K which depends on X (2) . Hence V+ (f, πn , (β, ))1 diverges as n → ∞ if γ(α − p2 ) + ((α − p2 ) − 1) > 0. If α > p2 we can find some γ > 0 large enough such that this is fulfilled and whenever p2 < 2 we are able to find an α with α > p2 such that there exists an α-stable process.  For any  ∈ (0, 1/2) the condition (2.68) is weaker than (2.9) as shown in the following remark. Remark 2.37. In the case p1 = p2 = 1 both conditions (2.68) and (2.69) become



(1)

|Ii,n |

1+(1/2−) 2

(2)

|Ij,n |

1+(1/2−) 2

1{I (1) ∩I (2) =∅} = OP (1). i,n

(1) (2) i,j:ti,n ∨tj,n ≤T

j,n

Hence the exponents of the observation interval lengths in the sum above both increase by 1/2 −  > 0 compared to the similar condition in Theorem 2.6. The improvement here is larger if  is smaller. The improvement in Theorem 2.35 yields that we get convergence of the functional V+ (f, πn , (β, ))T for the same class of functions as for V (f, πn )T but under a weaker condition. However, it might also be interpreted such that we get convergence for a wider class of functions under an identical condition. To illustrate this interpretation consider the case p = p1 = p2 < 2. Then the condition from Theorem 2.6 reads OP (1) =

 (1)

(2)

(1)



(1) (2) i,j:ti,n ∨tj,n ≤T

(2)

p

i,n

i,j:ti,n ∨tj,n ≤T

=

p

|Ii,n | 2 |Ij,n | 2 1{I (1) ∩I (2) =∅} (1)

|Ii,n |

p +(1/2−)(2−p ) 2

j,n

(2)

|Ij,n |

p+(1/2−)(2−p ) 2

1{I (1) ∩I (2) =∅} i,n

j,n

for p = (p − 2(1/2 − ))/(1/2 + ) < p. Hence under the above condition we get  the convergence of V+ (f, πn , (β, ))T for all functions f with f (x, y) = o(|xy|p ) as |xy| → 0 while we obtain convergence of V (f, πn )T only for functions f with f (x, y) = o(|xy|p ) as |xy| → 0. 

60

2 Laws of Large Numbers

For the normalized functionals the only improvement from using the truncated functionals compared to the ordinary ones is that X does not have to be continuous in the cases p = 2 respectively p1 + p2 = 2 such that we get convergence of (l)

V − (p, f, πn , (β, ))T , V − (p, g, πn , (β, ))T and V − (p1 + p2 , f, πn , (β, ))T in the settings of Theorem 2.17, Corollary 2.19 and Theorems 2.21, 2.22. This is the same improvement obtained in the setting of equidistant and synchronous (1) (2) observation times ti,n = ti,n = i/n; compare Theorem 9.2.1 in [30]. Theorem 2.38. Let β > 0 and  ∈ (0, 1/2). a) Suppose the assumptions made for Theorem 2.17 respectively Corollary 2.19 are fulfilled with the only difference that we have either p ∈ [0, 2] or p > 2 and X is continuous. Then it holds



P

V − (p, f, πn , (β, ))T −→

T

ρcs (f )dGp (s),

0

respectively (l) V − (p, g, πn , (β, ))T



P

−→ ρ1 (g)

T 0

(l)

(l)

(σs )p dGp (s),

l = 1, 2.

b) Suppose the assumptions made for Theorems 2.21 respectively Theorem 2.22 are fulfilled with the only difference that we have either p1 + p2 ∈ [0, 2] or p1 + p2 > 2 and X is continuous. Then V − (p1 + p2 , f, πn , (β, ))T converges to the limits in (2.41) respectively (2.42). Example 2.39. A special case in Theorem 2.22 was p1 = p2 = 1 where we obtained



V (2, f(1,1) , πn )T =

(1)

(2)

(1)

(2)

Δi,n X (1) Δj,n X (2) 1{I (1) ∩I (2) =∅} i,n

i,j:ti,n ∨tj,n P

−→



T 0

j,n

(1) (2)

ρs σs σs ds

for continuous processes X. Using truncated increments we then obtain by Theorem 2.38 via P

V − (2, f(1,1) , πn , (β, ))T −→



T 0

(1) (2)

ρs σs σs ds

a consistent estimator for the integrated co-volatility of X (1) and X (2) , which we will discuss in Chapter 7 in more detail, also for processes X with jumps. 

2.3 Functionals of Truncated Increments

61

Notes. In the textbooks [30] and [3] functionals of truncated increments are only considered in the setting of equidistant and synchronous observation times. [35] consider an estimator for the continuous part of the quadratic covariation of two processes X (1) and X (2) with finite jump activity based on asynchronous discrete observations which is of the form



(1)

(2)

Δi,n X (1) Δj,n X (2) 1{|Δ(1) X (1) |≤r(|π i,n

(1) (2) i,j:ti,n ∨tj,n ≤T

(2) (1) |≤r(|π | )} n |T ),|Δj,n X n T

× 1{I (1) ∩I (2) =∅} . i,n

(2.70)

j,n

Contrary to our approach they use a uniform threshold r(|πn |T ) which depends not on the length of the specific observation interval but on the mesh |πn |T of the observation scheme restricted to [0, T ]. Here, r : [0, ∞) → [0, ∞) denotes a function with r(x) → 0 sufficiently fast as x → 0. They find that the estimator (2.70) is T (1) (2) consistent for 0 ρs σs σs ds even for endogenous observation times provided that X has finite jump activity; compare Theorem 3.13 in [35]. Their result has been extended for processes of infinite jump activity and more general thresholds in [34].

2.3.2 The Proofs Proof of Theorem 2.33. This proof is inspired by the proof of Theorem 9.1.1 in [30]. Define fρ (x) = f (x)ψ(x/ρ) using a continuous function ψ : [0, ∞) → [0, 1] with ψ(u) = 0 for u ≤ 1/2 and ψ(u) = 1 for u ≥ 1 as in the proof of Theorem 2.3. Theorem 2.1 then yields P

V (fρ , πn )T −→ B(fρ )T as n → ∞ and similar arguments as in the proof of Lemma 2.14 yield the convergence B(fρ )T → B(f )T as ρ → 0. Hence it remains to prove





lim lim sup P 

ρ→0 n→∞





f (Δi,n X)1{Δi,n X>β|Ii,n | } − V (fρ , πn )T  > ε = 0

i:ti,n ≤T

(2.71) for all ε > 0. Therefore we compute

  

 i:ti,n ≤T

≤ Kρ

 

f (Δi,n X)1{Δi,n X>β|Ii,n | } − V (fρ , πn )T 



Δi,n X2 1{Δi,n X>β|Ii,n | } 1{Δi,n Xβ|Ii,n | } + 1{ρ/2≤Δi,n X β 2 |Ii,n |2 |S))1/2 ≤ K|Ii,n |

 E[Δ

i,n X|S β 2 |Ii,n |2

2

] 1/2

≤ K|Ii,n |1+(1/2−)

(2.77)

which is obtained using the Cauchy-Schwarz inequality, Markov’s inequality and (1.14), (1.16). The bound (2.76) vanishes as first n → ∞ and then q → ∞. Further the sum involving N (q) in (2.75) converges to



ΔN (q)s 2 1{ΔN (q)s  ε) = 0,

ρ→0 n→∞

(2.78) (2.79) ε > 0, (2.80)

2.3 Functionals of Truncated Increments

63

where fρ is defined as in the proof of Theorem 2.33. The convergence (2.79) directly follows from (2.21) because the limits are independent of the estimator. For showing (2.78) note that it holds





V+ (fρ , πn , (β, ))T − V (fρ , πn )T 1Ω(n,N,ρ)



=

(1)



(1)

(2)



fρ Δi,n X (1) , Δj,n X (2) 1{I (1) ∩I (2) =∅}

(2)

i,n

i,j:ti,n ∨tj,n ≤T

j,n

× 1{|Δ(1) X (1) |≤β|I (1) | ∨|Δ(2) X (2) |≤β|I (2) | } 1{|Δ(1) X (1) Δ(2) X (2) |>ρ/2} i,n

i,n

j,n

j,n

i,n

j,n

× 1Ω(n,N,ρ) =0 (l)

(l)

where Ω(n, N, ρ) denotes the set where |Δi,n X (l) | ≤ N holds whenver ti,n ≤ T , l = 1, 2, and where it holds β(|πn |T ) N ≤ ρ/2. Then (2.78) follows from lim lim sup P(Ω(n, N, ρ)) = 1

N →0 n→∞

for any ρ > 0 and Lemma 2.13. Finally we consider (2.80). In the setting from part a) the proof of (2.80) is identical to the proof of (2.22) because the estimate (2.23) also holds for |V+ (f, πn , (β, ))T − V+ (fρ , πn , (β, ))T |. For part b) note that we obtain using the same argument as in (2.27) |V+ (f, πn , (β, ))T − V+ (fρ , πn , (β, ))T |



≤ 2Kρ

 (1) (1) p1  (2) (2) p2  Δ X  Δ X  1 i,n

(1)

j,n

(2)

i,j:ti,n ∨tj,n ≤T

(1)

(2)

{Ii,n ∩Ij,n =∅}

× 1{|Δ(1) X (1) |>β|I (1) | ,|Δ(2) X (2) |>β|I (2) | } . i,n

i,n

j,n

(2.81)

j,n

Using the notation from (2.29) and arguments from the proof of Theorem 2.6 we obtain that the S-conditional expectation of the sum in (2.81) is bounded by Kp1 ,p2 T + Kp1 ,p2





(3−l) p3−l E |Δn,l,− X (l) |pl (|Δn | + |Δn,3−l,+ X (3−l) |p3−l ) kX k k

l=1,2 k:Tkn ≤T

× 1{|Δ(1)

(l) in (T n ),n k

+ Kp1 ,p2



X (1) |>β|I



(1) (3−l) (3−l) | ,|Δ (3−l) X (3−l) |>β|I (l) | } (l) in (T n ),n in (T n ),n in (T n ),n k k k

n,3−l,+ (3−l) p3−l (l) pl E |Δn X | k X | |Δk

l=1,2 k:Tkn ≤T

× 1{|Δ(1)

(l) in (T n ),n k

  S

X (1) |>β|I

(1) (3−l) (3−l) | ,|Δ (3−l) X (3−l) |>β|I (l) | } (l) in (T n ),n in (T n ),n in (T n ),n k k k

  S

64

2 Laws of Large Numbers

which in the case of p1 ∨ p2 < 2 is using H¨ older’s inequality for p = 2/(p1 ∨ p2 ), p = 2/(2 − p1 ∨ p2 ) bounded by 

Kp1 ,p2 T + Kp1 ,p2



2pl  E |Δn,l,− X (l) | p1 ∨p2 k

l=1,2 k:Tkn ≤T 2p3−l 2p3−l   (p ∨p )/2 (3−l) p ∨p × (|Δn | 1 2 + |Δn,3−l,+ X (3−l) | p1 ∨p2 )S 1 2 kX k

× E[1{|Δ(1)

(l) in (T n ),n k

+ Kp1 ,p2

(1)

(3−l)

(3−l)

X (1) |>β|I (l) | ,|Δ (3−l) X (3−l) |>β|I (l) | } in (T n ),n in (T n ),n in (T n ),n k k k  (2−p1 ∨p2 )/2 

...

... S 2p3−l   2pl   (p ∨p )/2 (l) p ∨p E |Δn | 1 2 |Δn,3−l,+ X (3−l) | p1 ∨p2 S 1 2 kX k



l=1,2 k:Tkn ≤T

× E[1{|Δ(1)

(l) in (T n ),n k

(1)

(3−l)

(3−l)

X (1) |>β|I (l) | ,|Δ (3−l) X (3−l) |>β|I (l) | } in (T n ),n in (T n ),n in (T n ),n k k k

...

 (2−p1 ∨p2 )/2 . . . S

≤ Kp1 ,p2 T +



+ Kp1 ,p2

(1)

(1)

|Ii,n |

p1 +(1/2−)(2−p1 ∨p2 ) 2

(2)

|Ij,n |

p2 +(1/2−)(2−p1 ∨p2 ) 2

(2)

i,j:ti,n ∨tj,n ≤T

× 1{I (1) ∩I (2) =∅} i,n

(2.82)

j,n

where we used iterated expectations, (1.12), the Cauchy-Schwarz inequality and the estimate for the indicator from (2.77) for the last inequality. In the case p1 ∨ p2 ≥ 2 we bound (2.81) by the same sum without the indicators and argue as in the proof of Theorem 2.6. Hence, (2.80) in the case (i) follows from (2.68) and Kρ → 0 for ρ → 0. Next we consider (2.80) in the case (ii). If X is of finite jump activity we have |V+ (f, πn (β, ))T − V+ (fρ , πn (β, ))T | ≤ 2Kρ



 (1) (1) p1  (2) (2) p2 Δ X  Δ X  1 i,n

(1)

j,n

(2)

i,j:ti,n ∨tj,n ≤T

(1)

(2)

{Ii,n ∩Ij,n =∅}

× 1{|Δ(1) X (1) |>β|I (1) | ,|Δ(2) X (2) |>β|I (2) | }

= 2Kρ



i,n

i,n

j,n

i,n

(1)

j,n

 (1)    Δ (B + C)(1) p1 Δ(2) (B + C)(2) p2 1 j,n

(2)

i,j:ti,n ∨tj,n ≤T

(1)

(2)

{Ii,n ∩Ij,n =∅}

× 1{|Δ(1) X (1) |>β|I (1) | ,|Δ(2) X (2) |>β|I (2) | } + Kρ OP (1) (2.83) i,n

i,n

j,n

j,n

2.3 Functionals of Truncated Increments

65

where

 Bt =

t 0

 bs −



R2

δ(s, z)1{δ(s,z)≤1} λ(dz) ds,

and the last identity in (2.83) holds because of Δ

(l) (l)

in (Sp ),n

X (l) → Δ

(l) (l)

in (Sp ),n

X (l) , l = 1, 2,

for any jump time Sp of X. The S-conditional expectation of (2.83) is using the Cauchy-Schwarz inequality and (2.77) bounded by



Kρ ((|πn |T )(p1 +p2 )/2 + 1)

(1)

(1)

|Ii,n |

p1 +(1/2−) 2

(2)

|Ij,n |

p2 +(1/2−) 2

(2)

i,j:ti,n ∨tj,n ≤T

× 1{I (1) ∩I (2) =∅} . i,n

j,n

(2.80) then follows from (2.69) as in case (i). Proof of Theorem 2.38. Comparing Proposition 2.30 and the proofs of Theorems 2.17, 2.21, 2.22 and Corollary 2.19 it suffices to prove ∗

∗

P

∗

∗

P

C − (p1 + p2 , f, πn , (β, ))T − C (p1 + p2 , f, πn )T −→ 0, p1 + p2 = 2, (2.84) V − (p1 + p2 , f, πn , (β, ))T − C (p1 + p2 , f, πn )T −→ 0, p1 + p2 = 2, (2.85) based on the assumptions made in Proposition 2.30 where we set ∗

C − (p, f, πn , (β, ))T = np/2−1



(1)

(1) (2) i,j≥0:ti,n ∨tj,n ≤T

(2)

(1) , Δ C (2) )1∗,n (i, j) f (Δi,n C j,n d1 ,d2

× 1{Δ(1) X  (1) ≤β|I (1) | ,Δ(2) X  (2) ≤β|I (2) | } ∗ V − (p, f, πn , (β, ))T

=n

p/2−1



(1)

i,n

(2)

i,n

j,n

j,n

(1)  (1) (2)  (2) ∗,n f (Δi,n X , Δj,n X )1d1 ,d2 (i, j)

i,j≥0:ti,n ∨tj,n ≤T

× 1{Δ(1) X  (1) ≤β|I (1) | ,Δ(2) X  (2) ≤β|I (2) | } i,n

i,n

j,n

j,n

 (l) is a zero-dimensional using the same notation as in (2.51). In the case that X (l)  (l) (l)  process we define Δi,n X  ≤ β|Ii,n | to be always true. Note that (2.84) is sufficient in the case p1 + p2 = 2 because ∗

∗

P

V − (p1 + p2 , f, πn , (β, ))T − C − (p1 + p2 , f, πn , (β, ))T −→ 0

66

2 Laws of Large Numbers

can be proven just like Proposition 2.30 as we may simply drop the indicators. For proving (2.84) we obtain using (2.45) and the estimate 1A∨B ≤ 1A + 1B for events A, B ∗

∗

E[C − (p, f, πn , (β, ))T − C (p, f, πn )T |S]   (l)  (l) pl ≤ np/2−1 K E[Δi,n C  l=1,2 i,j:t(l) ∨t(3−l) ≤T i,n

j,n

(3−l)  (3−l) p3−l × Δj,n C  1{Δ(l) C (l) >β|I (l) | } |S]1∗,n d1 ,d2 (i, j). i,n i,n

Using the Cauchy-Schwarz inequality, inequality (1.9), arguments as in (2.77) and the notation from (2.53) this quantity is bounded by np/2−1





l=1,2

(l) (3−l) i,j:ti,n ∨tj,n ≤T

(l)

(l) 4pl |S])1/4 K(E[Δi,n C

(l)

(l)

(3−l)

(l)  > β|I | |S)1/4 (E[Δ (3−l) 2p3−l |S])1/2 1∗,n (i, j) × (P(Δi,n C i,n j,n C d1 ,d2 (d ,d ),n

p11,p22 ≤ 2K(|πn |T )(1−2)/4 G

(T ).

Here the last bound vanishes as n → ∞ which proves (2.84) using Lemma 2.15. It remains to prove (2.85). To this end note that using inequality (2.46) we obtain the following estimate ∗

∗

E[V − (2, f, πn , (β, ))T − C (2, f, πn )T |S]  (1)  (1) p1 (2)  (2) p2 ≤ E[θ(ε)Δi,n C  Δj,n C  (1)

(2)

i,j:ti,n ∨tj,n ≤T

+





∗,n × 1{Δ(1) X  (1) ≤β|I (1) | ,Δ(2) X  (2) ≤β|I (2) | } |S]1d1 ,d2 (i, j) i,n

i,n

(l)  E[Kε Δi,n (X

j,n

j,n

 (l) pl − C)

l=1,2 i,j:t(l) ∨t(3−l) ≤T i,n

j,n

(3−l)

(3−l)

(3−l) p3−l + Δ   (3−l) p3−l ) × (Δj,n C j,n (X − C)

+



∗,n × 1{Δ(l) X  (l) ≤β|I (l) | ,Δ(3−l) X  (3−l) ≤β|I (3−l) | } |S]1d1 ,d2 (i, j)



i,n

l=1,2 i,j:t(l) ∨t(3−l) ≤T i,n

i,n

j,n

j,n

(1)  (1) p1 (2)  (2) p2 E[Δi,n C  Δj,n C  1{Δ(l) X  (l) >β|I (l) | } |S] i,n i,n

j,n

× 1∗,n d1 ,d2 (i, j).

2.4 Functionals of Increments over Multiple Observation Intervals

67

Let p1 , p2 > 0. Then the quantity above is using H¨ older’s inequality with exponents 2/p1 , 2/p2 , Lemma 1.4 and an argument similar to (2.77) bounded by (d ,d ),n

p11,p22 θ(ε)K G +



(T ) + Kε



(d ,d ),n

p11,p22 (Kq |πn |T )pl /2 + Keq )K G

(T )

l=1,2



(l)

 (q)(l) pl E[Δi,n N

l=1,2 i,j:t(l) ∨t(3−l) ≤T i,n

j,n

(3−l)

(3−l)

(3−l) p3−l + Δ   (3−l) p3−l )|S] × (Δj,n C j,n (X − C) ∗,n × 1{Δ(l) X  (l) ≤β|I (l) | ,Δ(3−l) X  (3−l) ≤β|I (3−l) | } |S]1d1 ,d2 (i, j) i,n

+ (|πn |T )

1/2−

i,n

j,n

j,n

p(d11,p,d22 ),n (T ) KG

where the sums in the first and last line vanish as first n → ∞ and then q → ∞ p(d11,p,d22 ),n (T ) = OP (1). Further the sum in the second line vanishes as because of G n → ∞ for any q > 0 using dominated convergence because we have (l)

(l)

(l)

 (q)(l)  =  (l)  > β|I |  0 ⇒ Δi,n X Δi,n N i,n P

with probability approaching one due to |πn |T −→ 0. Hence (2.85) follows using Lemma 2.15. For p1 = 0 or p2 = 0 the proof of (2.85) is based on (2.47). The arguments used in the case p1 = 0 and p2 = 0 are very similar to those in the case p1 , p2 > 0 and therefore not presented here.

2.4 Functionals of Increments over Multiple Observation Intervals In Chapter 8 and Section 9.2 we will use statistics where we compare functionals based on data sampled at different frequencies. Therefore for some k ≥ 2 we are interested in functionals which are of similar form as those discussed in Sections 2.1 and 2.2, but which are based on increments of X (1) and X (2) over the observation intervals (l)

(l)

(l)

Ii,k,n = (ti−k,n , ti,n ], l = 1, 2, (l)

(2.86)

(l)

instead of increments over Ii,n = Ii,1,n . We introduce the notation (l)

Δi,k,n X (l) = X

(l) (l)

ti,n

−X

(l) (l)

ti−k,n

(2.87)

68

2 Laws of Large Numbers (l)

(l)

and for convenience we again set Ii,k,n = ∅ and Δi,k,n X (l) = 0 if i < k. Then we 2

define for functions f : R → R, g : R → R the functionals



V (f, [k], πn )T =



(1) (2) i,j:ti,n ∨tj,n ≤T



V (l) (g, [k], πn )T =

(1)



(2)

f Δi,k,n X (1) , Δj,k,n X (2) 1{I (1)





(l)

(2) i,k,n ∩Ij,k,n =∅}

,

(2.88)

g Δi,k,n X (l) ,

l = 1, 2,

(l)

i:ti,n ≤T

and V (p, f, [k], πn )T =

V

(l)

np/2−1 kp/2+1

(p, g, [k], πn )T =





(1) (2) i,j:ti,n ∨tj,n ≤T

np/2−1 kp/2





(1)

[k],(l),n

(t) =

[k],n

Gp1 ,p2 (t) = [k],n

Hι,m,p (t) =

np/2−1 kp/2

,



(l)

(l)

i:ti,n ≤T

(l)

(p, g, [k], πn )T whenever they exist

  (l) p/2 I  , i,k,n

(l)

i:ti,n ≤t

n(p1 +p2 )/2−1 k(p1 +p2 )/2+1 np/2−1 kp/2+1

(2) i,k,n ∩Ij,k,n =∅}

g Δi,n X (l) , l = 1, 2.

To describe the limits of V (p, f, [k], πn )T , V we define the functions Gp



(2)

f Δi,k,n X (1) , Δj,k,n X (2) 1{I (1)



i,k,n

j,k,n

(1) (2) i,j:ti,n ∨tj,n ≤t

 (1)

 (1) p1 /2  (2) p2 /2 I  I  1

(1)

(2)

{Ii,k,n ∩Ij,k,n =∅}

,

 (1) (2) ι/2  (2) (1) m/2 I Ij,k,n \ Ii,k,n  i,k,n \ Ij,k,n

(2)

i,j:ti,n ∨tj,n ≤t



(1)

(2)

(p−(ι+m))/2

× Ii,k,n ∩ Ij,k,n 

1{I (1)

(2) i,k,n ∩Ij,k,n =∅}

.

(2.89) Here, we rescale interval lengths with the factor n/k such that the magnitude of the rescaled interval lengths is independent of k and n. Further we have an extra factor k−1 in V (p, f, [k], πn )T because we have approximately k times as many summands in V (p, f, [k], πn )T as in V (p, f, [1], πn )T . The fractions np/2−1 /kp/2+1 and np/2−1 /kp/2 in the definitions of V (p, f, [k], πn )T and V

(l)

(l)

(p, g, [k], πn )T are

chosen such that the magnitude of V (p, f, [k], πn )T and V (p, g, [k], πn )T does not depend on k at least for equidistant and synchronous observation times (1) (2) ti,n = ti,n = i/n.

2.4 Functionals of increments over multiple observation intervals

69

2.4.1 The Results The results we obtain for the asymptotic behaviour of the functionals based on (l) increments over the intervals Ii,k,n are very similar to the results obtained in Sections 2.1 and 2.2. As a generalization of Corollary 2.2, Theorem 2.3 and Theorem 2.6 we obtain the following result for the non-normalized functionals. Theorem 2.40. Suppose Condition 1.3 is fulfilled. a) It holds P

V (l) (g, [k], πn )T −→ kB (l) (g)T . for all continuous functions g : R → R with g(x) = o(x2 ) as x → 0. b) It holds P

V (f, [k], πn )T −→ k2 B(f )T

(2.90)

for all continuous functions f : R → R with f (x, y) = O(x2 y 2 ) as |xy| → 0. c) Let p1 , p2 > 0 with p1 + p2 ≥ 0. We have (2.90) for all continuous functions f : R → R with f (x, y) = o(|x|p1 |y|p2 ) as |xy| → 0 if the observation scheme is exogenous and



 (1)  p1 ∧1  (2)  p2 ∧1 I  2 I 2 1 i,k,n

(1)

j,k,n

(2)

i,j:ti,n ∨tj,n ≤t

(1)

(2)

{Ii,k,n ∩Ij,k,n =∅}

= OP (1). (2.91)

For the normalized functionals we get the following results where Theorem 2.41 contains the results in the setting of synchronous observation times and Theorem 2.42 the results in the setting of asynchronous observation times. Theorem 2.41. Suppose Condition 1.3 is fulfilled and the observation scheme is exogenous. Further assume that it either holds p ∈ [0, 2) or we have p ≥ 2 and X is continuous. a) For all positively homogeneous functions g : R → R of degree p it holds V

(l)



P

(p, g, [k], πn )T −→ ρ1 (g)

T 0

(l)

[k],(l)

(σs )p dGp

(s) [k],(l)

if there exists some (possibly random) continuous function Gp [0, ∞) with [k],(l),n

Gp

P

[k],(l)

(t) −→ Gp

(t),

t ∈ [0, T ].

: [0, T ] →

70

2 Laws of Large Numbers b) Further if the observation scheme is synchronous and if there exists some [k] (possibly random) continuous function Gp : [0, T ] → [0, ∞) with [k],(1),n

Gp

P

[k]

(t) −→ Gp (t),

then it holds P

V (p, f, [k], πn )T −→



t ∈ [0, T ],

T

[k]

ρcs (f )dGp (s)

0

for all positively homogeneous functions f : R2 → R of degree p. Theorem 2.42. Suppose Condition 1.3 is fulfilled and the observation scheme is exogenous. Further assume that either p1 , p2 ≥ 0 with p1 + p2 ∈ [0, 2) or we have p1 + p2 ≥ 2 and X is continuous. a) Suppose ρ ≡ 0 on [0, T ] and assume that P

[k],n

[k]

Gp1 ,p2 (t) −→ Gp1 ,p2 (t), t ∈ [0, T ], [k]

holds for some (possibly random) continuous function Gp1 ,p2 : [0, T ] → [0, ∞). Then for all continuous functions f : R2 → R which are positively homogeneous of degree p1 in the first component and positively homogeneous of degree p2 in the second component it holds



P

V (p1 + p2 , f, [k], πn )T −→ ρI2 (f )

T 0

(1)

(2)

[k]

(σs )p1 (σs )p2 dGp1 ,p2 (s).

Here I2 denotes the two-dimensional identity matrix. [k],n

b) Let p1 , p2 be non-negative integers and assume that Gp1 ,p2 (T ) = OP (1) holds. Further assume that for all ι, m ∈ N0 for which an l ∈ N0 exists with (ι, l, m) ∈ L(p1 , p2 ), for the notation compare Theorem 2.22, there exist [k] (possibly random) continuous functions Hι,m,p1 +p2 : [0, T ] → [0, ∞) which fulfill P

[k],n

[k]

Hι,m,p1 +p2 (t) −→ Hι,m,p1 +p2 (t),

t ∈ [0, T ].

Then it holds V (p1 + p2 , f(p1 ,p2 ) , [k], πn )T P

−→



T 0

(1) (2) (σs )p1 (σs )p2



 (ι,l,m)∈L(p1 ,p2 )

× ρ1 (x

p1 +p2 −(ι+l+m)

where f(p1 ,p2 ) (x, y) = xp1 y p2 .



  p1 ι, l

p2 ρ1 (xι )ρ1 (xl )ρ1 (xm ) m [k]



)(1 − ρ2s )l/2 (ρs )p1 −(ι+l) dHι,m,p1 +p2 (s) .

2.4 Functionals of increments over multiple observation intervals

71

Example 2.43. The assumptions from Theorems 2.40–2.42 are fulfilled in the setting of Poisson sampling; compare Definition 5.1. Indeed that the assumption [k],(l),n [k],n [k],n (2.91) is fullfilled and that the functions Gp , Gp1 ,p2 , Hι,m,p converge in probability to linear deterministic functions follows directly from Corollary 5.8. 

2.4.2 The Proofs Proof of Theorem 2.40. It holds V (l) (g, [k], πn )T =

k 



(l)

g(Δik+ι,k,n X (l) )

ι=1 i:t(l)

ik+ι,n ≤T

where each of the sums



(l)

g(Δik+ι,k,n X (l) ),

ι = 0, . . . , k − 1,

(l)

i:tik+ι,n ≤T

converges to B (l) (g)T by Corollary 2.2. Therefore note that (X

(l) (l) tι,n

−X

(l)

(l) (l) t0,n



)+

(l)

ι g(Δik+ι,k,n X (l) ) = V (l) (g, πn )T

(l)

i:tik+ι,n ≤T

(l)

ι with πn = {(0, tk+ι,n t2k+ι,n , . . .)|l = 1, 2} and

(X

(l) (l)

tι,n

−X

(l) (l)

t0,n

) = oP (1),

ι = 0, . . . k − 1,

as n → ∞. Similarly we proof parts b) and c) using V (f, [k], πn )T =

k 



(1)

(2)

f (Δik+ι,k,n X (1) , Δ(jk+ι ,k,n X (2) )

ι,ι =1 i,j:t(1)

(2) ik+ι,n ∨tjk+ι ,n ≤T

× 1{I (1)

(2) ik+ι,k,n ∩Ijk+ι,k,n =∅}

and Theorems 2.3, 2.6. For part c) note that (2.8) is fulfilled for each (ι, ι ) because of (2.91). Proof of Theorems 2.41 and 2.42. The proofs are identical to the proofs in Section 2.2.

3 Central Limit Theorems In this chapter, we will discuss central limit theorems for some of the functionals introduced in Chapter 2. Further we develop general techniques that will later in Chapters 7–9 be applied to find central limit theorems also for other more specific statistics which are based on functionals from Chapter 2. Throughout this chapter we will assume that the observation scheme is exogenous. In practice a theory for endogenous observation times might be desirable as well. However, previous research shows that for such observation schemes it is difficult to derive central limit theorems even in simple situations; compare [18] or [47]. For this reason we restrict ourselves to exogenous observation times. This setting is already far more general than the setting of equidistant and synchronous observation times (1) (2) ti,n = ti,n = i/n which is mainly discussed in the literature. It covers a lot of interesting random and irregular sampling schemes where the most prominent one is Poisson sampling; see Definition 5.1. Already in the setting of exogenous observation times the resulting central limit theorems have a more complicated structure as the limit not only depends on the process X but also on the asymptotics of the observation scheme. Hence we will need to impose further assumptions that guarantee an appropriate behaviour of the observation schemes πn for n → ∞. As common in this field of high-frequency statistics the asymptotic variances in the upcoming central limit theorems are themselves random. They e.g. depend via the volatility process σ and jump heights ΔXs , s ∈ [0, T ], on the observed path of the process X. For this reason we derive stable central limit theorems as is done frequently in related literature. This means that we do not look for convergence in law but stable convergence in law of the properly rescaled error term to a certain random limit. Stable convergence has the advantage that a consistent estimator for the random asymptotic variance can be used to construct asymptotically exact confidence intervals. To illustrate this property let us consider the stable convergence √

L−s

n(Yn − Y ) −→ V U

for random variables Y , (Yn )n∈N , V and U ∼ N (0, 1) where V is not independent of Y, (Yn )n∈N . If we now can find a consistent sequence of estimators (Vn )n∈N P

with Vn −→ V we also obtain

√

n(Yn − Y ) L−s −→ U. Vn

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 O. Martin, High-Frequency Statistics with Asynchronous and Irregular Data, Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics, https://doi.org/10.1007/978-3-658-28418-3_3

74

3 Central Limit Theorems

This implication in general does not hold if we consider ordinary convergence in law instead of stable convergence in law. In particular the above implication holds in general for ordinary convergence in law only if V is deterministic. A more detailed introduction of the concept of stable convergence in law and the discussion of some of its properties which are used in the upcoming proofs can be found in Appendix B.

3.1 Central Limit Theorem for Non-Normalized Functionals In this section we will derive central limit theorems for the non-normalized functionals V (l) (g, πn )T and V (f, πn )T . We will first sketch the results using so-called toy examples. These are models for the process X (l) respectively X which have the simplest form among all processes which yield a central limit theorem of the general form i.e. they contain all relevant terms that contribute in the limit but were freed of terms that do not. We start with motivating the result in the univariate setting because in that situation the structure of the result is simpler. To this end we consider toy,(l)

Xt



t

= 0

(l)

(l)

σs dWs +



ΔN (l) (q)s ,

t ≥ 0,

s≤t

for some q > 0 where the process X toy,(l) consists of a Brownian part with volatility process σ (l) and a finite activity jump part. Finite jump activity means toy,(l) that Xt almost surely has only finitely many jumps in any interval [0, t], t ≥ 0. Further let g : R → R denote a twice continuously differentiable function with t (l) toy,(l) (l) g(0) = g (0) = 0 and g

(x) = o(|x|) as x → 0. We denote Ct = 0 σs dWs . By means of a Taylor expansion we then obtain the following identity

 √  (l) n V (g, πn )T − B (l) (g)T    √ (l) (l) (l) = n g(Δi,n N (l) (q)) + g (Δi,n N (l) (q))Δi,n C toy,(l) (l)

i:ti,n ≤T

+

   1

(l) (l) (l) (l) g(ΔN (l) (q)s ) g (Δi,n N (q) + θi,n Δi,n C toy,(l) )(Δi,n C toy,(l) )2 − 2 s≤T

(3.1) (l)

for some (random) θi,n ∈ [0, 1]. Denote by Sp , p ∈ N, an enumeration of the (countably many) jump times of N (l) (q) on [0, ∞). Conditional on the set where

3.1 Central Limit Theorem for Non-Normalized Functionals

75

two jumps of N (l) (q) in [0, T ] are further apart than |πn |T the expression (3.1) is due to g(0) = g (0) = 0 equal to √



n

g (ΔN (l) (q)S (l) )Δ p

(l) p:Sp ≤T

√

n 2

+



(l) (l)

(l)

in (Sp ),n

C toy,(l)

(l)

(3.2)

(l)

(l)

g

(Δi,n N (l) (q) + θi,n Δi,n C toy,(l) )(Δi,n C toy,(l) )2 . (3.3)

(l)

i:ti,n ≤T

For the sum (3.3) we obtain by the assumption g

(x) = o(|x|) √



n 2

(l)

(l)

(l)

g

(Δi,n N (l) (q) + θi,n Δi,n C toy,(l) )(Δi,n C toy,(l) )2

(l)

i:ti,n ≤T

√ =

n 2



g

(ΔN (l) (q)S (l) + θi(l) (S (l) ),n Δ p

(l)

n

p

p:Sp ≤T

× (Δ + oP

√

(l) (l)



n

(l)

in (Sp ),n

|Δ

(l)

i:ti,n ≤T

(l) (l)

(l)

in (Sp ),n

C toy,(l) )

C toy,(l) )2

(3.4)



(l) (l)

(l)

in (Sp ),n

C toy,(l) |3 .

(3.5)

Here, the sum (3.4) contains almost surely only finitely many summands of which √ (l) each is of order n|I (l) (l) |. Hence (3.4) vanishes for n → ∞ because we will assume |I

in (Sp ),n

(l) (l)

(l)

in (Sp ),n

E

√

n

| = OP (1/n). Further it holds by inequality (1.9)



|Δ

(l)

i:ti,n ≤T

 

(l) (l)

(l)

in (Sp ),n

(l),n

C toy,(l) |3 S ≤ KG3

(T ).

(l),n

Hence (3.5) also vanishes as n → ∞ if we assume G3 (T ) = OP (1) as n → ∞. Combining these observations we obtain that (3.3) is asymptotically negligible by the assumption on g. For the sum (3.2) we obtain the approximation √

n



g (ΔN (l) (q)S (l) )Δ

(l) p:Sp ≤T

× (σ

(l) (l)

Sp −

(l)

L

(l) (l)

(l)

p

in (Sp ),n

(l)

(l)

C toy,(l) ≈

(l)

(nδn,− (Sp ))1/2 U− (Sp ) + σ



g (ΔN (l) (q)S (l) ) p

(l) p:Sp ≤T

(l) (l)

Sp

(l)

(l)

(l)

(l)

(nδn,+ (Sp ))1/2 U+ (Sp ))

76

3 Central Limit Theorems (l)

(l)

δn,− (s)

δn,+ (s)

X (l) t

 (l) I (l)

(l) (l)

in (s)−1,n

in (s),n

 

t

(l) (l)

in (s),n

s (l)

(l)

Figure 3.1: Illustration of the terms δn,− (s) and δn,+ (s). (l)

(l)

where U− (s), U+ (s), s ∈ [0, T ], are i.i.d. standard normal random variables independent of N (l) (q) and σ (l) and the random variables (l)



(l)

(δn,− (s), δn,+ (s)) = s − t

(l) (l) in (s)−1,n

,t



(l) (l) in (s),n

− s , s ≥ 0,

(3.6)

describe the distances of s to previous and upcoming observation times; compare Figure 3.1. (l) (l) If we assume that the random variables (nδn,− (s), nδn,+ (s)), s ∈ [0, T ], converge (l)

(l)

in a suitable sense to random variables (δ− (s), δ+ (s)), s ∈ [0, T ], we obtain the stable convergence



 L−s √  (l) n V (g, πn )T − B (l) (g)T −→

g (ΔN (l) (q)S (l) ) p

(l)

p:Sp ≤T

× (σ

(l)

(l)

(l)

Sp −

(l)

(l)

(l)

(l)

(l)

(δ− (Sp ))1/2 U− (Sp ) + σ (l)

(l) (l)

Sp

(l)

(l)

(l)

(l)

(l)

(δ+ (Sp ))1/2 U+ (Sp )) (l)

(l)

(l)

(3.7)

(l)

where (δ− (Sp ), δ+ (Sp )) is independent of U− (Sp ), U+ (Sp ). Here, the limit has a mixed normal distribution with random variance





(g (ΔN (l) (q)S (l) ))2 (σ p

(l)

p:Sp ≤T

(l)

(l)

(l)

Sp −

(l)

)2 δ− (Sp ) + (σ

(l) (l)

Sp

(l)

)2 δ+ (Sp )



which depends on N (l) (q) and σ (l) as well as on the asymptotics of the observation scheme. Next, we will motivate the result in the bivariate setting with asynchronous observation times. We again consider a toy example

 Xttoy =

t 0

σs dWs +

 s≤t

ΔN (q)s ,

t ≥ 0,

3.1 Central Limit Theorem for Non-Normalized Functionals

77

which is of similar form as the one in the univariate setting. Let f : R2 → R be a twice continuously differentiable function with f (x, y) = ∂1 f (x, y) = ∂2 f (x, y) = 0 for all (x, y) ∈ R2 with xy = 0, ∂kl f (x, y) = o(|x| (k,l)

(k,l)

p1

|y|

(k,l)

p2

(k,l)

) as |xy| → 0

(k,l)

(3.8) (3.9)

(k,l)

for some p1 , p2 ≥ 0 with p1 + p2 = 1 for any k, l ∈ {1, 2}. Here ∂l denotes the first order partial derivative with respect to the l-th argument and ∂kl denotes the second order partial derivative where we first differentiate with respect to the l-th argument and then with respect to the k-th argument. By Taylor expansion, compare Theorem 7.2 in [17], we obtain

 √  n V (f, πn )T − B ∗ (f )T   √  (1) (2) = n f (Δi1 ,n N (1) (q), Δi2 ,n N (2) (q)) (l) (2) ∨ti ,n ≤T 1 ,n 2

i1 ,i2 :ti



+

(1)

(2)

(l)

∂l f (Δi1 ,n N (1) (q), Δi2 ,n N (2) (q))Δil ,n C toy,(l)

l=1,2

 1

+

2

k,l=1,2

(1)

(1)

(2)



−

(2)

∂kl f (Δi1 ,n N (1) (q) + θin1 ,i2 Δi1 ,n C (1) , Δi2 ,n N (2) (q) (k)

(l)

+ θin1 ,i2 Δi2 ,n C (2) )Δik ,n C toy,(k) Δil ,n C toy,(l) f (ΔN (1) (q)s , ΔN (2) (q)s )





(3.10)

s≤T

for some θin1 ,i2 ∈ [0, 1]. Hence on the set where any two different jump times of N (q) in [0, T ] are further apart than 2|πn |T the expression (3.10) is using (3.8) equal to √

n





∂l f (Δ

p:Sp ≤T l=1,2

(1)

in (Sp ),n

N (1) (q), Δ

(2) (2)

in (Sp ),n

N (2) (q))Δ

(l) (1)

in (Sp ),n

C toy,(l) (3.11)

√

n + 2

(1)

 (l) (2) ∨ti ,n ≤T 1 ,n 2

i1 ,i2 :ti

(2)





(1)

(1)

∂kl f Δi1 ,n N (1) (q) + θin1 ,i2 Δi1 ,n C (1) ,

k,l=1,2 (2)



(k)

(l)

Δi2 ,n N (2) (q) + θin1 ,i2 Δi2 ,n C (2) Δik ,n C toy,(k) Δil ,n C toy,(l)

(3.12)

where Sp , p ∈ N, denotes an enumeration of the common jump times of N (1) (q) and N (2) (q).

78

3 Central Limit Theorems Splitting the sum (3.12) into the summands where (1)

(2)

(Δi1 ,n N (1) (q), Δi2 ,n N (2) (q)) = 0 and into those summands where increments of N (q) do not contribute similarly as in the univariate setting for (3.4) and (3.5) we obtain that (3.12) is equal to 2 

oP (1) +



oP Gn(k,l) p1

k,l=1

(k,l)

+(2−k)+(2−l),p2

+(k−1)+(l−1)

(T )



which vanishes as n → ∞ if we assume that Gn(k,l) p1

(k,l)

+(2−k)+(2−l),p2

(T ), k, l = 1, 2,

+(k−1)+(l−1)

are all OP (1) as n → ∞. Hence (3.12) is asymptotically negligible. As in the univariate setting the quantity (3.11) constitutes the terms which are relevant in the limit. It holds   √ (1) (2) (l) n ∂l f (Δ (1) N (1) (q), Δ (2) N (2) (q))Δ (l) C toy,(l) in (Sp ),n

p:Sp ≤T l=1,2 L

≈

 

p:Sp ≤T

in (Sp ),n



(1)

∂1 f (ΔN (1) (q)Sp , ΔN (2) (q)Sp ) σSp − (1)

+ σ Sp

$

+





(1),+

Rn (Sp )USp

(1)

(1)

(1)

(1)

(2)

(σSp − )2 Ln (Sp ) + (σSp )2 Rn (Sp )USp



(2)

+ σ Sp ρSp

$

+ (1),−

(1),−

Ln (Sp )USp

(2)

+ ∂2 f (ΔN (1) (q)Sp , ΔN (2) (q)Sp ) σSp − ρSp −

+

in (Sp ),n





 (1),−

Ln (Sp )USp

(1),+

Rn (Sp )USp

(2)

(2)

(3)

(σSp − )2 (1 − (ρSp − )2 )Ln (Sp ) + (σSp )2 (1 − (ρSp )2 )Rn (Sp )USp

$

(2)

(2)

(2)

(2)

(4)

(σSp − )2 Ln (Sp ) + (σSp )2 Rn (Sp )USp

(1),+

(2)

(3)



(3.13)

(4)

where Us , Us , Us , Us , Us , s ∈ [0, T ], are i.i.d. standard normal distributed and we use the random variables (l)

Ln (s) = max{t (l)

Rn (s) = t

(1) (1)

in (s)−1,n

(l) (l)

in (s),n

− min{t

Ln (s) = s − max{t Rn (s) = min{t

(2) (2)

in (s)−1,n

(1) (1)

in (s),n

(1) (1) in (s)−1,n

(1) (1)

,t

in (s),n

,t

,t

(l)

in (s)−1,n

(2)

in (s),n

(2) in (s)−1,n

in (s),n

(l)

(2)

(2)

(2) (2)

,t

}−t

}−s

},

},

,

l = 1, 2,

l = 1, 2, (3.14)

3.1 Central Limit Theorem for Non-Normalized Functionals (1)

Ln (s) = 0

Ln (s)

79 (1)

Rn (s)

Rn (s)

X (1)  (2) I

(2)

in (s),n

 (1) I

 

(1)

in (s),n

 

X (2) (2)

Ln (s)

Ln (s)

Rn (s)

(2)

Rn (s) = 0

s (l)

(l)

Figure 3.2: Illustration of the terms Ln (s), Rn (s), l = 1, 2, and Ln (s), Rn (s). to describe lengths of overlapping and non-overlapping parts of the observation intervals containing s; compare Figure 3.2. As in the univariate situation we then obtain convergence of (3.11) and hence √ also of n(V (f, πn )T − B ∗ (f )T ) if we assume that the random vectors (1)

(1)

(2)

(2)

(nLn (s), nRn (s), nLn (s), nRn (s), nLn (s), nRn (s)), s ∈ [0, T ], (3.15) converge in a suitable sense to random variables (L(1) (s), R(1) (s), L(2) (s), R(2) (s), L(s), R(s)), s ∈ [0, T ].

(3.16)

The limit is of the form (3.13) with the only difference that the rescaled interval lengths (3.15) are replaced with their limits (3.16). Again we observe that the limit has a mixed normal distribution with random variance. If we move from our toy examplse with finite jump activity to general Itˆ o semimartingales X we obtain the limit variables in the central limit theorems for general X by letting q → ∞ in (3.7) and (3.15). To show that the stable convergence is preserved if we take the limit q → ∞ and to formally justify the approximations and estimates sketched above we will need a lot of technical arguments in Section 3.1.2. However, all relevant ideas that lead to the form of the limits in the general central limit theorems can be illustrated using our toy examples.

3.1.1 The Results In this section we state precise central limit theorems for the non-normalized functionals V (l) (g, πn )T in the univariate setting and V (f, πn )T in the bivariate setting based on the previous considerations for our toy examples. As for the

80

3 Central Limit Theorems

motivation we start by presenting the result in the univariate setting. The necessary assumptions are summarized in the following condition. Condition 3.1. Let Condition 1.3 be fulfilled and the observation scheme be √ exogenous. Further it holds n|πn |T = oP (1) and (l),n

(i) we have G3

(l),n

(T ) = OP (1), G4

(T ) = OP (1) as n → ∞.

(ii) The integral



T 0

g(x1 , . . . , xP )E

P



(l)



(l)

hp (nδn,− (xp ), nδn,+ (xp )) dx1 . . . dxP (3.17)

p=1

converges as n → ∞ to



P  

T 0

g(x1 , . . . , xP )

p=1

R

hp (y)Γuniv,(l) (xp , dy)dx1 . . . dxP

(3.18)

for all bounded functions g : [0, T ]P → R, hp : [0, ∞)2 → R, p = 1, . . . , P , and any P ∈ N. Here Γuniv,(l) (x, dy), x ∈ [0, T ], is a family of probability measures on [0, ∞)2 with uniformly bounded first moments. (l)

(l)

Let Z univ,(l) (s) = (δ− (s), δ+ (s)), s ∈ [0, T ], be random variables defined on

) whose distribution is given by  F, P an extended probability space (Ω, Z P (l)

univ,(l)

(x)

(dy) = Γuniv,(l) (x, dy).

(l)

Further let (U− (s), U+ (s)), s ∈ [0, T ], denote i.i.d. standard normal distributed

) as well. The random variables  F, P random variables which are defined on (Ω, (l) (l) univ,(l) Z (s) are independent of the (U− (s), U+ (s)). All the random variables (l)

(l)

Z univ,(l) (s), (U− (s), U+ (s)) are independent of each other for different values of s ∈ [0, T ] and independent of the σ-algebra F which contains the information on the process X and its components. Theorem 3.2. Suppose Condition 3.1 is fulfilled. Then we have the X -stable convergence

 √  (l) n V (g, πn )T − B (l) (g)T L−s

−→



(l)

(l)

(l)

(l)

(l)

(l)

(l)

g (ΔXs )(σs− (δ− (s))1/2 U− (s) + σs (δ+ (s))1/2 U+ (s))

(3.19)

s≤T

for all functions g : R → R with g(0) = g (0) = 0 and g

(x) = o(|x|) as x → 0.

3.1 Central Limit Theorem for Non-Normalized Functionals

81

Here, the notion that a sequence of F -measurable real-valued random variables (Yn )n∈N converges X -stably in law to a random variable Y defined on an extended ) means that it holds  F, P probability space (Ω,

 [g(Y )Z] E[g(Yn )Z] −→ E for all bounded functions g : R → R and all bounded real-valued X -measurable random variables Z. For more details on stable converge in law consult Appendix B. Considering the functions gp (x) = xp and g p (x) = |x|p for p ≥ 0 introduced in Section 2.2 we observe that we get convergence only for p > 3. Hence to obtain a central limit theorem we need to impose a stricter condition than was needed to P

get mere convergence V (l) (g, πn )T −→ B (l) (g)T where only p > 2 was required; compare Corollary 2.2. The assumptions on the function g in the univariate irregular setting are identical to the assumptions which were made in Theorem 5.1.2 of [30] to obtain a central limit theorem in the case of equidistant deterministic observation times. Condition 3.1(ii) is a necessary assumption in the setting of (l) (l) irregular observation schemes because we need that (δn,− (s), δn,+ (s)) converges in law as motivated by the toy example. Condition 3.1(i) is an additional structural assumption on the moments of the observation intervals. It is automatically fulfilled in the setting of equidistant observations which is shown in Example 3.3. In the following examples we discuss Condition 3.1 and Theorem 3.2 for two important observation schemes. The first example shows that our results are a natural generalization of the existing results in the setting of equidistant observation times. In fact we will see that the result from Theorem 3.2 reduces to Theorem 5.1.2 in [30] in this specific setting. In the second example we discuss the genuinely random observation scheme of Poisson sampling and show that our results can be applied in this setting as well. Example 3.3. Condition 3.1 is fulfilled in the setting of equidistant observation (l) times ti,n = i/n. Indeed in that case it holds (l),n Gp (T )

nT 

=n

p/2−1



n−p/2 → T, p ≥ 0.

i=1

Further the outer integral in (3.17) can be interpreted as the expectation with regard (l) to P independent uniformly distributed random variables Sp on [0, T ]. Then, as (l)

(l)

(l)

(l)

the variables (nδn,− (Sp ), nδn,+ (Sp )) are asymptotically distributed like (κ, 1 − κ) (l)

(l)

with κ ∼ U [0, 1], we obtain the limiting distribution of (nδn,− (s), nδn,+ (s)) as (l)

(l)

(δ− (s), δ+ (s)) ∼ (κ(l) (s), 1 − κ(l) (s)) for independent κ(l) (s) ∼ U[0, 1]. Standard

82

3 Central Limit Theorems

arguments which will be formalized in the proof of Lemma 5.10 further show (l) (l) (l) (l) (l) (l) (l) (l) that (nδn,− (Sp ), nδn,+ (Sp )) and (nδn,− (Sp ), nδn,+ (Sp )) are asymptotically

independent for p = p . Hence Condition 3.1 is fulfilled in the setting of equidistant observation times and we obtain from Theorem 3.2

 √  (l) n V (g, πn )T − B (l) (g)T L−s

−→



(l)

(l)

(l)

(l)

(l)

g (ΔXs )(σs− (κ(l) (s))1/2 U− (s) + σs (1 − κ(l) (s))1/2 U+ (s)).

s≤T

Here the limit is identical to the limit in Theorem 5.1.2 of [30] which we recover for univariate processes from Theorem 3.2 in the special case of equidistant observations.

 Example 3.4. Condition 3.1 is also fulfilled in the setting of Poisson sampling (l) (l) where the observation times ti,n are recursively defined by t0,n = 0 and (l)

(l)

(l)

ti,n = ti−1,n + Ei,n /n (l)

for i.i.d. Exp(λl )-distributed random variables Ei,n for some fixed λl > 0. Indeed Condition 3.1(i) holds by Corollary 5.6 and that part (ii) holds follows from Lemma 5.10 with f (l) (x, y, z) = (x, y). (l) By the memorylessness of the exponential distribution the variable nδn,+ (s) is Exp(λl )-distributed and by the discussion following Lemma 5.10 the backward (l) waiting time nδn,− (s) is asymptotically Exp(λl )-distributed as well and independent (l)

(l)

(l)

(l)

(l)

of nδn,+ (s). Hence we get (δ− (s), δ+ (s)) ∼ (E− (s), E+ (s)) for i.i.d. Exp(λl )(l)

(l)

distributed random variables E− (s), E+ (s), s ∈ [0, T ], and (3.19) becomes

 √  (l) n V (g, πn )T − B (l) (g)T L−s

−→



(l)

(l)

(l)

(l)

(l)

(l)

(l)

g (ΔXs )(σs− (E− (s))1/2 U− (s) + σs (E+ (s))1/2 U+ (s))

s≤T



in the case of Poisson sampling.

Next, we present the central limit theorem for V (f, πn )T in the setting of asynchronous and random observation schemes. The following condition summarizes the necessary assumptions. We use the notation (1)

(1)

(2)

(2)

Znbiv (s) = (nLn (s), nRn (s), nLn (s), nRn (s), nLn (s), nRn (s)). (3.20) Condition 3.5. Let Condition 1.3 be fulfilled and the observation scheme be √ exogenous. Additionally, it holds n|πn |T = oP (1) and

3.1 Central Limit Theorem for Non-Normalized Functionals

83

n (i) it holds Gn 3,0 (T ) = OP (1), G0,3 (T ) = OP (1) as n → ∞.

(ii) The integral



T 0

g(x1 , . . . , xP )E

P





hp (Znbiv (xp )) dx1 . . . dxP

p=1

converges as n → ∞ to



T 0

g(x1 , . . . , xP )

P   p=1

R6

hp (y)Γbiv (xp , dy)dx1 . . . dxP

for all bounded functions g : [0, T ]P → R, hp : [0, ∞)6 → R, p = 1, . . . , P , and any P ∈ N. Here Γbiv (x, dy), x ∈ [0, T ], is a family of probability measures on [0, ∞)6 with uniformly bounded first moments. As for the result in the univariate setting we need an extended probability space ) on which the limit can be defined. On (Ω, ) we define via  F,  P  F, P (Ω, Z biv (s) = (L(1) (s), R(1) , L(2) (s), R(2) , L(s), R(s)), s ∈ [0, T ],

Z (x) (dy) = Γbiv (x, dy). random variables which are distributed according to P (1),− (1),+ (2) (3) (4) Further let Us , Us , Us , Us , Us be i.i.d. standard normal distributed ) as well. All these newly introduced random  F, P random variables defined on (Ω, variables should be independent of each other and independent of F . Using these variables we define biv

Φbiv T (f ) =



(1)

(2)



(1)

∂1 f (ΔXs , ΔXs ) σs−

s≤T

$ +

(1)



(1)

(1)

(2)



(2)

$

+

$

(2)

(2)



(1)

+ σs

(σs− )2 L(1) (s) + (σs )2 R(1) (s)Us

+ ∂2 f (ΔXs , ΔXs ) σs− ρs− +

(1),−

L(s)Us

(1),−

L(s)Us



(1),+

R(s)Us

 (2)

+ σ s ρs

(2)



(1),+

R(s)Us (3)

(σs− )2 (1 − (ρs− )2 )L(s) + (σs )2 (1 − (ρs )2 )R(s)Us (2)

(2)

(4)

(σs− )2 L(2) (s) + (σs )2 R(2) (s)Us



Theorem 3.6. If Condition 3.5 is fulfilled we have the X -stable convergence

 L−s √  n V (f, πn )T − B ∗ (f )T −→ Φbiv T (f ) for all functions f : R2 → R which fulfil (3.8)–(3.9).

(3.21)

84

3 Central Limit Theorems

In the following we discuss the conditions (3.8)–(3.9) imposed on the function f to obtain a central limit theorem for V (f, πn )T . The assumption f (x, y) = 0 whenever xy = 0 was already needed in Section 2.1 to obtain convergence of V (f, πn )T where the limit solely consisted of common jumps. To obtain a central limit theorem we additionally require that the first derivatives fulfil ∂1 f (x, y) = ∂2 f (x, y) = 0 whenever xy = 0. Hence the limit Φbiv T (f ) depends solely on the common jumps of X (1) and X (2) as well. That the condition on the first derivatives is necessary becomes obvious when looking at (3.10). There in the sum √



n



(l) (2) ∨ti ,n ≤T 1 ,n 2

i1 ,i2 :ti

(1)

(2)

(l)

∂l f (Δi1 ,n N (1) (q), Δi2 ,n N (2) (q))Δil ,n C toy,(l) (3.22)

l=1,2

for an idiosyncratic jump of N (1) (q) at time Sp asymptotically the term √ (l) ∂l f (ΔN (1) (q)Sp , 0) nΔ (1)

in (Sp ),n

would be included in (3.22) exactly |{j : I

C toy,(1)

(l) (1)

in (Sp ),n

(2)

∩ Ij,n = ∅}| times. How-

ever, this number in general does not converge. Hence because of the property √ (l) nΔ (1) C toy,(1) = OP (1) we need ∂l f (ΔN (1) (q)Sp , 0) = 0 to ensure converin (Sp ),n

gence of (3.22). The condition (3.9) on the second derivatives is e.g. fulfilled for all functions f (p1 ,p2 ) (x, y) = |x|p1 |y|p2 with p1 ∧ p2 > 2. This requirement corresponds to the condition made on the function f in Theorem 2.3. As in the univariate setting we discuss Condition 3.5 in the special setting of equidistant synchronous observation times and in the case of Poisson sampling. Example 3.7. Condition 3.5 is fulfilled in the setting of equidistant observation (l) (2) times ti,n = ti,n = i/n. In fact it holds (1),n

Gn p1 ,p2 (T ) = Gp1 +p2 (T ), p1 , p2 ≥ 0, (1)

(1)

Ln = δn,− , Rn = δn,+ , (1)

(2)

(1)

(2)

Ln = Ln = Rn = Rn = 0, due to the synchronicity of the observation times. Hence that Condition 3.5 is fulfilled in the equidistant setting can be shown as in Example 3.3. Similarly as for (3.19) in Example 3.3 the convergence in (3.21) in this setting is identical to the corresponding statement for bivariate processes in Theorem 5.1.2 of [30]. 

3.1 Central Limit Theorem for Non-Normalized Functionals

85

Example 3.8. Condition 3.5 is also fulfilled in the setting of Poisson sampling. √ The assumption n|πn |T = oP (1) is fulfilled by (5.2). Further Condition 3.5(i) is fulfilled by Corollary 5.6 and 3.5(ii) follows from Lemma 5.10 where f (3) is chosen to be the projection on the first 3 arguments. In the case of Poisson samling the distribution Γbiv (s, dy) can be characterized (l) (l) as follows: Let E− (s), E+ (s), s ∈ [0, T ], be i.i.d. Exp(λl )-distributed random variables for l = 1, 2. Then the limit distribution of nZnbiv (s) is given by Z biv (s) = (L(1) (s), R(1) , L(2) (s), R(2) , L(s), R) (1)

(2)

(1)

(2)

L(s) = E− (s) ∧ E− (s), R(s) = E+ (s) ∧ E+ (s),



(l)

(1)

(2)

L(l) (s) = E− (s) − E− (s) ∧ E− (s) R

(l)

(s) =

(1)



+

,

+ (l) (1) (2) E+ (s) − E+ (s) ∧ E+ (s) (2)

(1)

(2)

using L(s) = δ− (s) ∧ δ− (s), R(s) = δ− (s) ∧ δ− (s), the considerations in Example 3.4 and the independence of the two Poisson processes.  Notes. The techniques used in this section are based on methods from [8]. In particular Conditions 3.1(ii) and 3.5(ii) are analogous to Assumption 2(ii) and Assumption 4(ii) in [8]. Further, the limiting variables (δ− (s), δ+ (s)) and Z biv (s) are also very similar to those in Theorems 2 and 3 of [8]. Our limiting variables are slightly more complicated because we allow for discontinuous volatility processes σs .

3.1.2 The Proofs Proof of Theorem 3.2. For the sake of clarity we first present the rough structure of the proof while the more technical computations necessary for the individual steps are presented later. Step 1. First we discretize σ (l) and restrict ourselves to the big jumps ΔN (l) (q)s of which there are almost surely only finitely many. We define σ (l) (r), C (l) (r) for (l)

r ∈ N by σ (l) (r)t = σ(j−1)2−r if t ∈ [(j −1)2−r , j2−r ), C (l) (r)t = Here, (1)

W t = (Wt (l)

(1)

, ρt Wt

$ +

(2) ∗

1 − ρ2t Wt

)

t 0

(l)

σ (l) (r)s dW s .

(3.23)

is defined such that W denotes for l = 1, 2 the Brownian motion driving the (l) process X (l) . Denote by Sq,p , p ∈ N, an enumeration of the jump times of N (l) (q). By x = min{n ∈ Z : n ≥ x} we denote for x ∈ R the smallest integer which is

86

3 Central Limit Theorems

greater or equal to x. Using the notion of the jump times we modify the discretized processes σ (l) (r), C (l) (r) further via

 σ ˜

(l)

(r, q)s =

(l) (r, q)t = C

σ σ



(l)

t 0

(l)

(l)

if s ∈ [Sq,p , 2r Sq,p /2r )

(l)

Sq,p (l)

(r)s

,

otherwise (l)

σ ˜ (l) (r, q)s dW s .

Using this notation we then define R(l) (n, q, r) =

√



n

(l)

(l)

(l) (r, q). g (Δi,n N (l) (q))Δi,n C

i:ti,n ≤T

and show √ lim lim sup lim sup P(| n(V (l) (g, πn )T − B (l) (g)T ) − R(l) (n, q, r)| > ε) = 0

r→∞ q→∞

n→∞

(3.24) for all ε > 0. The proof of (3.24) will require some very technical estimates and will be given after the discussion of the rough structure of the proof. Step 2. Next we will show the X -stable convergence L−s

R(l) (n, q, r) −→ ΦT

univ,(l)



(g, q, r) :=

g (ΔN (l) (q)S (l) ) q,p

(l)

Sq,p ≤T



(l)

(l)

(l)

× σ ˜ (l) (r, q)S (l) − (δ− (Sq,p ))1/2 U− (Sq,p ) q,p

(l)

(l)

(l)

+σ ˜ (l) (r, q)S (l) (δ+ (Sq,p(l) ))1/2 U+ (Sq,p )



(3.25)

q,p

for all q > 0 and r ∈ N. The proof of (3.25) is based on Condition 3.1(ii) and will also be given after the discussion of the rough structure of the proof. Step 3. Finally we show

(|Φ lim lim sup P T

univ,(l)

q→∞ r→∞

univ,(l)

(g) − ΦT

(g, q, r)| > ε) = 0,

(3.26)

for all ε > 0 where univ,(l)

ΦT

(g) =

 s≤T

(l)

(l)

(l)

(l)

(l)

(l)

(l)

g (ΔXs )(σs− (δ− (s))1/2 U− (s) + σs (δ+ (s))1/2 U+ (s))

3.1 Central Limit Theorem for Non-Normalized Functionals

87

denotes the limit in (3.19). To show (3.26) note that it holds univ,(l)

|ΦT =



univ,(l)

(g) − ΦT

|g (ΔM

(l)

(g, q, r)| (l)

(l)

(l)

(l)

(l)

(l)

(q)s )||σs− (δ− (s))1/2 U− (s) + σs (δ+ (s))1/2 U+ (s)|

(3.27)

s≤T

+



|g (ΔN (l) (q)S (l) )|(|σ q,p

(l)

Sq,p ≤T

+ |σ

(l)

(l)

(l)

(l)

Sq,p −

(l)

(l)

(l)

−σ ˜ (l) (r, q)S (l) − |(δ− (Sq,p ))1/2 |U− (Sq,p )| q,p

(l)

(l)

(l)

(l)

−σ ˜ (l) (r, q)S (l) |(δ+ (Sq,p ))1/2 |U+ (Sq,p )|). (3.28)

(l)

Sq,p

q,p

Using the boundedness of σ and that the measures Γuniv,(l) (·, dy) have uniformly bounded first moments we obtain that the F -conditional expectation of (3.27) is bounded by K



|g (ΔM (l) (q)s )|.

(3.29)

s≤T

Note that g (0) = 0 and g

(x) = o(|x|) imply g (x) = o(|x|2 ) as x → 0. Hence the sum (3.29) is almost surely finite for any q and therefore vanishes as q → ∞. This observation yields using Lemma 2.15 that (3.27) also vanishes in probability as q → ∞. Further (3.28) vanises almost surely as r → ∞ for any q > 0 because σ is (l) (l) c` adl` ag and because of σ ˜ (l) (r, q)S (l) = σ (l) for any Sq,p . Hence we have proven Sq,p

q,p

(3.26). Step 4. Combining the results (3.24), (3.25) and (3.26) from steps 1–3 we obtain the claim (3.19) using Lemma B.6. Lemma 3.9. Let Condition 1.3 be satisfied, the processes σt , Γt be bounded and the observation scheme be exogenous. Then there exists a constant K which is independent of (i, j) such that

 (l) 2  (l ) 2  E Δi,n C Δj,n M (q) σ(F

(l )

(l)

ti,n ∧tj,n





(l)  (l ) 

, S) ≤ Keq Ii,n Ij,n ,

l, l ∈ {1, 2}. (3.30)

(l )

(l)

Proof. If Ii,n ∩ Ij,n = ∅ we use iterated expectations and (1.9), (1.10). If the intervals do overlap, we use iterated expectations for the non-overlapping parts to obtain

 (l) 2  (l ) 2  E Δi,n C Δj,n M (q) σ(F



+E C

(l) (l) (l ) ti,n ∧tj,n



−C

(l )

(l)

ti,n ∧tj,n













 

2

(l) (l) (l ) ti−1,n ∨tj−1,n

× M (3−l) (q)



(l) (l ) (l) (l ) 2 , S) ≤ Keq Ii,n Ij,n  − Ii,n ∩ Ij,n 



(l)

(l )

ti,n ∧tj,n

− M (l ) (q)

(l)

(l )

ti−1,n ∨tj−1,n

2  σ(F

(l)

(l )

ti,n ∧tj,n



, S) .

88

3 Central Limit Theorems (l )

(l)

The claim now follows from (8.16) in [32] which is basically (3.30) for Ii,n = Ij,n . Proof of (3.24). To prove (3.24) we first split √ | n(V (l) (g, πn )T − B (l) (g)T ) − R(l) (n, q, r)| into multiple terms which we will then discuss separately. To this end note that on the set Ω(l) (n, q) where any two different jumps of N (l) (q) in [0, T ] are further apart than |πn |T it holds √ | n(V (l) (g, πn )T − B (l) (g)T ) − R(l) (n, q, r)|1Ω(l) (n,q) √    (l) (l) g(Δ (l) (l) X (l) ) − g(Δ (l) (l) (X (l) − N (l) (q))) ≤ n in (Sq,p ),n

(l)

Sq,p ≤T

√  + n

in (Sq,p ),n

 



− g(ΔN (l) (q)S (l) ) − R(l) (n, q, r) (3.31)



(l) g(Δi,n (X (l)

−N

(l)

(q))) −

(l)



 

q,p

g(ΔM (l) (q)s ).

(3.32)

s≤T

i:ti,n ≤T

Hence because of P(Ω(l) (n, q)) → 1 as n → ∞ for any q > 0 it remains to show that (3.31) and (3.32) vanish in the sense of (3.24). Step 1. We first discuss (3.31). Using a Taylor expansion we obtain g(Δ

(l) (l)

(l)

in (Sq,p ),n

X (l) ) = g(ΔN (l) (q)S (l) ) + g (ΔN (l) (q)S (l) )Δ q,p

q,p

(l) (l)

(l)

in (Sq,p ),n

(X (l) − N (l) (q))

 (l) 2 1 (l) n + g

(ΔN (l) (q)S (l) + θq,p Δ (l) (l) (X (l) − N (l) (q))) Δ (l) (l) (X (l) − N (l) (q)) q,p in (Sq,p ),n in (Sq,p ),n 2 n for some random θq,p ∈ [0, 1]. Hence it remains to show

lim lim sup P(|Y1 (n, q, r)| + |Y2 (n, q, r)| + |Y3 (n, q, r)| > ε) = 0

r→∞ n→∞

(3.33)

for all ε > 0 and any q > 0 where 

√ g (ΔN (l) (q)S (l) ) Y1 (n, q, r) = n q,p

(l)

Sq,p ≤T

×Δ Y2 (n, q, r) =

√

n

√ Y3 (n, q, r) =

n 2



(l) (l)

g(Δ

(l) Sq,p ≤T



(l)

in (Sq,p ),n

(l) (r, q)), (B (l) (q) + M (l) (q) + C (l) − C

(l) (l)

(l)

in (Sq,p ),n

(X (l) − N (l) (q))),

n g

(ΔN (l) (q)S (l) + θq,p Δ q,p

(l)

Sq,p ≤T



× Δ

(l) (l) (l) in (Sq,p ),n

(l) (l)

(l)

in (Sq,p ),n

2

(X (l) − N (l) (q)) .

(X (l) − N (l) (q)))

3.1 Central Limit Theorem for Non-Normalized Functionals

89

The key ingredient in the proof of (3.33) is to show √ (l) lim lim sup P( n|Δ (l)

(l)

in (Sq,p ),n

r→∞ n→∞

(l) (r, q))| > ε) = 0 (B (l) (q) + M (l) (q) + C (l) − C (3.34) (l)

for all ε > 0 and any jump time Sq,p . (3.34) can be proven similarly as (4.4.23) in (l)

[30]. To this end note that a filtration Gt (q) can be defined such that the Sq,p are G0 (q)-measurable and the processes B (l) (q), M (l) (q), C (l) , C (l) (q, r) have the same properties as Gt (q)-adapted processes like they have as Ft -adapted processes. Then (l) Sq,p is in particular Gt(l) (q)-measurable and estimates like in Lemma 1.4 −1 (l) (l) in (Sq,p ),n

can be used. By inequalities (1.8) and (2.1.34), (2.1.39) from [30] it then can be shown that lim lim sup E

r→∞ n→∞

√

n|Δ

(l) (l)

(l)

in (Sq,p ),n

(l) (r, q))| (B (l) (q) + C (l) − C 

√ (l) + min{ n|Δ (l)

(l) in (Sq,p ),n



M (l) (q))|, c}σ(G0 (q) ∪ S) = 0, c ≥ 0,

holds. The proof of this convergence makes use of the fact that Condition 3.1(ii) (l) (l) (l) (l) implies the X -stable convergence of (nδn,− (Sq,p ), nδn,+ (Sq,p )) which will be shown in the proof of (3.25). For more details consult the arguments leading up to (4.4.23) in [30]. A slightly more formal construction of the filtration (Gt (q))t≥0 will be presented in the proof of Corollary 6.4. Using (3.34) we obtain that Y1 (n, q, r) vanishes as first n → ∞ and then r → ∞ for any q > 0 because N (l) (q) almost surely only has finitely many jumps in [0, T ]. (l) Further on the set Ω(l) (n, m, q, T ) where |Δ (l) (l) (X (l) − N (l) (q))| < 1/m for any

(l) Sq,p

in (Sq,p ),n

≤ T we obtain



|Y2 (n, q, r)|1Ω(l) (n,m,q,T ) ≤ n−1

(l)

Sq,p ≤T

√ (l) Km | nΔ (l)

(l)

in (Sq,p ),n

(X (l) − N (l) (q))|3

because g(0) = g (0) = 0 and g

(x) = o(|x|) imply g(x) = o(|x|3 ) as x → ∞. (3.34) √ (l) √ (l) then shows that nΔ (l) (l) (X (l) − N (l) (q)) and nΔ (l) (l) C (l) (q, r) are in (Sq,p ),n

in (Sq,p ),n

asymptotically equivalent and in the proof of (3.25) we will see that the sequence √ (l) nΔ (l) (l) C (l) (q, r) converges stably in law which yields in (Sq,p ),n

√

nΔ

(l) (l)

(l)

in (Sq,p ),n

C (l) (q, r) = OP (1).

Hence Y2 (n, q, r) vanishes as n → ∞ and then r → ∞ for any q > 0 because of limn→∞ P(Ω(l) (n, m, q, T )) = 1 for any m > 0 and q > 0.

90

3 Central Limit Theorems

On the set Ω

(l)

(n, M, q, T ) where

|g

(ΔN (l) (q)S (l) + θΔ q,p

(l) (l)

(l)

in (Sq,p ),n

(X (l) − N (l) (q)))| ≤ M

(l)

for any θ ∈ [0, 1] and all Sq,p ≤ T we obtain |Y3 (n, q, r)|1

Ω

(l)

(n,m,q,T )

≤

n−1/2 2



√ (l) M | nΔ (l)

(l)

in (Sq,p ),n

(l)

Sq,p ≤T

(X (l) − N (l) (q))|2

and hence Y3 (n, q, r) can be discussed similarly as Y2 (n, q, r) because of lim lim sup P(Ω(n, M, q, T )) = 1.

M →∞ n→∞

Combining the above discussions of Yi (n, q, r), i = 1, 2, 3, we have proved (3.33). Step 2. Next we consider (3.32). Using again a Taylor expansion it remains to prove lim lim sup P(|Y4 (n, q)| + |Y5 (n, q)| + |Y6 (n, q)| > ε) = 0

q→∞ n→∞

for any ε > 0 where Y4 (n, q) =

√  n



(l)

g(Δi,n M (l) (q)) −

(l)

Y5 (n, q) =

√

n



g(ΔM (l) (q)s ) ,

s≤T

i:ti,n ≤T





(l)

(l)

g (Δi,n M (l) (q))Δi,n (B (l) (q) + C (l) ),

(l)

√

n 2

Y6 (n, q) =

i:ti,n ≤T



(l)

(l)

g

(Δi,n M (l) (q) + θin Δi,n (B (l) (q) + C (l) ))

(l) i:ti,n ≤T



(l)

× Δi,n (B (l) (q) + C (l) )

2

for random variables θin ∈ [0, 1]. Using Itˆ o’s formula, compare Theorem 3.21 in [41], for the processes



M (l) (q)t − M (l) (q)t(l)

i−1,n

 (l)

t≥ti−1,n

, i, n ∈ N,

(3.35)

3.1 Central Limit Theorem for Non-Normalized Functionals

91

we obtain that Y4 (n, q) is equal to √  n







g(0) +

(l)

ti−1,n

(l)

i:ti,n ≤T

+



1 2

g (M (l) (q)s− − M (l) (q)t(l)

g

(M (l) (q)s− − M (l) (q)t(l)

(l)

)d[M (l) (q), M (l) (q)]cs

i−1,n



(l)

g(M (l) (q)s − M (l) (q)t(l)

i−1,n

(l)

s∈(ti−1,n ,ti,n ]

− g (M (l) (q)s − M (l) (q)t(l)

) − g(M (l) (q)s− − M (l) (q)t(l)

+

√



n

√



)ΔM (l) (q)s



(l)



g(ΔM (l) (q)s )



(l) i:ti,n ≤T

(l) (l) s∈(ti−1,n ,ti,n ]

)dM (l) (q)s

g(M (l) (q)s − M (l) (q)t(l)

)

i−1,n

)

i−1,n

− g(ΔM (l) (q)s ) − g (M (l) (q)s − M (l) (q)t(l) √



)ΔM (l) (q)s .

(3.37)

i−1,n



n

(3.36)

i−1,n

− g(M (l) (q)s− − M (l) (q)t(l)

+



s≤T

g (M (l) (q)s− − M (l) (q)t(l)



n

−

(l)

ti,n ti−1,n

(l) i:ti,n ≤T

)

i−1,n

i−1,n

=

)dM (l) (q)s

i−1,n

(l)

ti,n ti−1,n

+

(l)

ti,n

g(ΔM (l) (q)s )

(3.38)

(l) s∈(t (l) ,T ] in (T ),n

where [Y, Y ]cs denotes the part of the quadratic variation [Y, Y ]s originating from the continous part of a process Y . For Y = M (l) (q) it holds [M (l) (q), M (l) (q)]cs ≡ 0. Following the ideas used in Step 4) in the proof of Theorem 5.1.2 in [30] we set (l) T (l) (n, m, i) = inf{s > ti−1,n : |M (l) (q)s − M (l) (q)t(l) | > 1/m} and i−1,n

(l)

(l)

% (l) (n, m, q, T ) = {T (l) (n, m, i) ≥ t for all i with t ≤ T }. Ω i,n i,n % (l) (n, m, q, T ) (3.36) is identical to Then on the set Ω √

n

 (l)

i:ti,n ≤T



(l)

ti,n ∧T (n,m,i) (l)

ti−1,n

g (M (l) (q)s− − M (l) (q)t(l)

)dM (l) (q)s . (3.39)

i−1,n

Using that (3.39) is a sum of martingale differences and the Burkholder-DavisGundy inequality (A.1) the S-conditional expectation of the square of (3.36) is,

92

3 Central Limit Theorems

compare (29) in [8], bounded by (by δ (l) (s, z) we denote the l-th component of δ(s, z) ∈ R2 )

(l)     ti,n ∧T (n,m,i)

E (g (M (l) (q)s− − M (l) (q)t(l) ))2 d[M (l) (q), M (l) (q)]s S n (l)

≤n



E





Keq E





Keq E



(l)





))2

i−1,n

(l)

ti,n ∧T (n,m,i) (l)

i:ti,n ≤T

≤n

(g (M (l) (q)s− − M (l) (q)t(l)

Keq



(l) ti−1,n

(l),n

≤ Km e q G 4

Km |M (l) (q)s− − M (l) (q)t(l)

i−1,n

  |4 dsS

(l)

ti,n (l)

(l)

  ))2 dsS

(l)

ti,n ∧T (n,m,i)

ti−1,n

i:ti,n ≤T

(g (M (l) (q)s− − M (l) (q)t(l)

i−1,n

ti−1,n

(l)



R2

  × (δ (l) (s, z))2 1{γ(z)≤1/q} λ(dz)dsS

i:ti,n ≤T

≤n



(l)

(l)



(l)

ti,n ∧T (n,m,i) ti−1,n

i:ti,n ≤T

≤n

i−1,n

ti−1,n

(l)

i:ti,n ≤T

Km E[|M (l) (q)s− − M (l) (q)t(l)

i−1,n

|4 |S]ds

(T )

where we used g (x) = o(|x|2 ) as x → 0 such that Km → 0 as m → 0 and inequality (1.10). Hence we have shown that (3.36) vanishes as first n → ∞ and then q → ∞ % (l) (n, m, q, T )) = 1 for any m > 0. because of limq→∞ lim supn→∞ P(Ω Further we obtain from (5.1.22) in [30] |g(x + y) − g(x) − g(y) − g (x)y| ≤ Km |x||y|2

% (l) (n, m, q, T ) intersected with for x, y ∈ R with |x|, |y| ≤ 1/m. Hence on the set Ω the set where no jump of M (q) in [0, T ] is larger than 1/m we obtain that the S-conditional expectation of the absolute value of (3.37) is bounded by    

√ E n Km |M (l) (q)s− − M (l) (q)t(l) ||ΔM (l) (q)s |2 S (l)

(l)

i−1,n

(l)

i:ti,n ≤T s∈(ti−1,n ,ti,n ]

√ = E Km n





√

= E Km n

 (l) i:ti,n ≤T



(l) ti−1,n

(l) i:ti,n ≤T

(l)

ti,n

R2

|M (l) (q)s− − M (l) (q)t(l)

i−1,n

|

 



(l)

ti,n (l)

ti−1,n

× (δ (l) (s, z))2 1{γ(z)≤1/q} μ(dz, ds)S

 R2

|M (l) (q)s− − M (l) (q)t(l)

i−1,n

|

3.1 Central Limit Theorem for Non-Normalized Functionals

93

 





√ ≤ Km n

Keq

× (δ (l) (s, z))2 1{γ(z)≤1/q} λ(dz)dsS (l)

ti,n (l)

ti−1,n

(l) i:ti,n ≤T

E[|M (l) (q)s− − M (l) (q)t(l)

i−1,n

||S]ds.

Here the S-conditional expectation of the expression in the last line is using (1.10) (l),n less or equal to Km eq G3 (T ). Hence (3.37) also vanishes as first n → ∞ and then q → ∞ by Condition 3.1(i). Finally the S-conditional expectation of the √ sum (3.38) is using inequality (1.10) bounded by nKeq |πn |T which vanishes by √ the assumption n|πn |T = oP (1) and we have shown all together that Y4 (n, q) is asymptotically negligible. % (l) (n, m, q, T ) we obtain using the CauchyNext we discuss Y5 (n, q). On the set Ω Schwarz inequality, inequalities (1.8)–(1.10) and Lemma 3.9

E[|Y5 (n, q)|1Ω  (l) (n,m,q,T ) |S]  √ (l) (l) ≤ n Km E[(Δi,n M (l) (q))2 |Δi,n (B (l) (q) + C (l) )||S] (l)

i:ti,n ≤T

√ ≤ Km n





(l)

E[(Δi,n M (l) (q))2 |S]

(l)

i:ti,n ≤T (l)

(l)

× E[(Δi,n M (l) (q))2 (Δi,n (B (l) (q) + C (l) ))2 |S] (l),n

≤ Km eq (Kq (|πn |T )1/2 + 1)G3

1/2

(T )

where the expression in the last line vanishes as n → ∞ and then q → ∞ by Condition 3.1(i). Hence Y5 (n, q) is asymptotically negligible by Lemma 2.15 and % (l) (n, m, q, T )) = 1 as well. because of limq→∞ lim supn→∞ P(Ω (l)  (n, m, q, T ) the set where Finally denote by Ω (l)

(l)

Δi,n M (l) (q)s + θΔi,n (B (l) (q) + C (l) ) < 1/m (l)

for all i with ti,n ≤ T and all θ ∈ [0, 1]. On this set |Y6 (n, q)| is bounded by √

n 2





(l)

(l)

(l)

i:ti,n ≤T

√ ≤

(l)

Km |Δi,n M (l) (q)s + θin Δi,n (B (l) (q) + C (l) )| Δi,n (B (l) (q) + C (l) )

n 2





(l)

(l)

Km |Δi,n M (l) (q)s | + |Δi,n (B (l) (q) + C (l) )|

(l)

i:ti,n ≤T



(l)

× Δi,n (B (l) (q) + C (l) )

2



2

94

3 Central Limit Theorems

where the S-conditional expectation of the expression in the last line is using (l),n the Cauchy-Schwarz inequality and (1.8)–(1.10) bounded by Km G3 (T ) which vanishes as first n → ∞ and then m → ∞. Hence we have shown (3.35). Proof of (3.25). Step 1. We define the process A(l) (q)t =

 t R2

0

1{γ(z)>1/q} μ(ds, dz), t ≥ 0,

(l)

and denote by Sq,p , p ∈ N, an enumeration of its jump times, compare (4.3.1) in [30]. Further we set



(l)

(l)

(l)

(l)

(l)

∗

Yn (s) = nδn,− (s), nδn,+ (s), Un,− (s), Un,+ (s) ,



(l)

(l)

(l)

∗

(l)

Y (l) (s) = δ− (s), δ+ (s), U− (s), U+ (s) , with (l)

(l)

Un,− (s) = (W s − W (l)

Un,+ (s) = (W

(l)

(l) t

(l)

(l) (l) in (s)−1,n

(l)

(l) t (l) in (s),n

)/(δn,− (s))1/2 (3.40)

(l)

− W s )/(δn,+ (s))1/2 .

where W t is defined as in (3.23). We begin by showing that Condition 3.1(ii) yields the X -stable convergence of (l) (l) (l) all the Yn (Sq,p ) to the respective Y (l) (Sq,p ), i.e. we have to show



 (l) (l)    (l)   Λf Y (l) (Sq,p E Λf Yn (Sq,p ) S(l) ≤T → E ) S(l) ≤T q,p

(3.41)

q,p

for all X -measurable bounded random variables Λ and all bounded Lipschitz functions f . We will use the same techniques to prove this convergence as were used in [8] e.g. in the proof of Proposition 3. Similar arguments can also be found in the proof of Lemma 6.2 in [29] and in the proof of Lemma 5.8 in [27]. Denote by Ω(l) (q, m, n) the subset of Ω on which |πn |T < 1/m and where two (l) different jumps Sq,p ≤ T are further apart than |πn |T . As P(Ω(l) (q, m, n)) → 1 for n → ∞ it suffices to prove (3.41) with the indicator 1Ω(l) (q,m,n) added in both expectations. Further we set

&

B (l) (m) =



(l)

(l)

(l)

q,p ≤T S

W

(l)



(m)t =

t 0



max{Sq,p − 1/m, 0}, min{Sq,p + 1/m, T } ,

1B (l) (m) (s)dW

(l)

(s).

3.1 Central Limit Theorem for Non-Normalized Functionals

95

(l)

(l)

Let G(m) denote the σ-algebra generated by W (m) and the jump times Sq,p ≤ T . By conditioning on σ(G(m) ∪ S) we see that for proving (3.41) with the indicator 1Ω(l) (q,m,n) added in both expectations it is sufficient to consider only (l)

G(m)-measurable Λ , as restricted to Ω(l) (q, m, n) the Yn (Sp ) are σ(G(m) ∪ S)measurable. By Lemma 2.1 in [29] we may in particular choose Λ of the form Λ = γ(W

(l)



(l)

(m))κ (Sq,p )S(l) ≤T



q,p

for bounded Lipschitz functions γ and κ. Because W obtain

(l)

(m) converges to 0 as m → ∞ and because γ, κ, f are bounded we

 



 

(l)

lim lim sup E γ(0)κ (Sq,p )S(l) ≤T f

m→∞ n→∞

− E γ(W

(l)

q,p



(l)

 

(m))κ (Sq,p )S(l) ≤T f q,p

(l)

(l)

(l)

Yn (Sq,p ) (l)

(l)

Yn (Sq,p )





 (l)

q,p ≤T S



(l) q,p S ≤T

(l)

 1Ω(l) (q,m,n)  = 0

(l)

and the analogous result for Yn (Sq,p ) replaced with Y (l) (Sq,p ). Hence it remains to prove

 (l)   (l) (l)   E κ (Sq,p )S(l) ≤T f Yn (Sq,p ) S(l) ≤T q,p q,p

 (l)  κ (Sq,p →E ) (l)

 

Sq,p ≤T

f

(l)

Y (l) (Sq,p )



 (l)

q,p ≤T S

(3.42)

for all bounded Lipschitz functions κ, f . Further note that by another density argument, compare again Lemma 2.1 in [30], it suffices to consider functions f of the form f



(l)

(l)

Yn (Sq,p )



 (l)

q,p ≤T S



=

(l) 

fp







(l) (l) (l) (l) (l) (l) (l) (l) (l) nδn,− (Sq,p ), nδn,+ (Sq,p ) f˜p Un,− (Sq,p ), Un,+ (Sq,p ) .

(l)

p:Sq,p ≤T

(l)

(l)

(l)

(l)

Then because the Un,− (Sq,p ), Un,+ (Sq,p ) are i.i.d. N (0, 1)-distributed and inde(l)

(l)

(l)

(l)

pendent of μ and (nδn,− (Sq,p ), nδn,+ (Sq,p )), (3.42) becomes

 (l)   (l)  (l) (l) (l) (l)  E κ (Sq,p )S(l) ≤T fp nδn,− (Sq,p ), nδn,+ (Sq,p ) q,p

(l)

q,p ≤T S



(l)

 κ (Sq,p ) (l) →E  ≤T S q,p





(l)  (l) (l) (l) (l)  δ− (Sq,p ), δ+ (Sq,p ) .

fp

(3.43)

(l) q,p p:S ≤T

However, this is exactly Condition 3.1(ii) as conditional on the event that there (l) are P jumps of A(l) (q) in [0, T ] all the Sq,p are independent uniformly distributed

96

3 Central Limit Theorems

on [0, T ] by the properties of the Poisson random measure μ, compare Section 1.3.1 [3]. This argument occurs in similar form in [8], compare the paragraph following (42) in that paper. Note that the second expectation in (3.43) can be written in (l) (l) (l) the form (3.18) as the (δ− (s), δ+ (s)) are independent of the Sq,p and of each other. Hence we have shown



(l)

(l)

Yn (Sq,p )

 (l)

q,p ≤T S

 L−s  −→

(l)

Y (l) (Sq,p )



 (l)

q,p ≤T S

.

(3.44)

Step 2. Denote by Ω(l) (q, r, n) the subset of Ω where two different jump times (l) (l) = Sq,p of N (l) (q) are further apart than |πn |T and the jump times Sq,p are r further away than |πn |T from the discontinuity points k/2 of σ ˜ (q, r). On this set we get (l) Sq,p

R(l) (n, q, r)1Ω(l) (q,r,n)



=

(l)



(l)

(l)

(l)

(l)

g (ΔN (l) (q)S (l) ) σ ˜ (r, q)S (l) − (nδn,− (Sq,p ))1/2 Un,− (Sq,p ) q,p

q,p

Sq,p ≤T (l)

(l)

(l)

(l)



+σ ˜ (r, q)S (l) (nδn,+ (Sq,p ))1/2 Un,+ (Sq,p ) 1Ω(l) (q,r,n) . (3.45) q,p

On the other hand from (3.44) and Lemma B.3 we obtain





(l) 

N (q), σ ˜ (q, r), Sq,p



(l)

q,p ≤T S

L−s

(l)

(l)

, Yn (Sq,p )





 (l)

q,p ≤T S



(l) −→ N (q), σ ˜ (q, r), Sq,p





(l)

q,p ≤T S

(l)

, Y (l) (Sq,p )



 (l)

q,p ≤T S

which using the continuous mapping theorem for stable convergence in law stated in Lemma B.5 yields

 (l)



(l)

(l)

(l)

(l)

g (ΔN (l) (q)S (l) ) σ ˜ (r, q)S (l) − (nδn,− (Sq,p ))1/2 Un,− (Sq,p ) q,p

q,p

Sq,p ≤T (l)

(l)

(l)

(l)

 L−s

(g, q, r). +σ ˜ (r, q)S (l) (nδn,+ (Sq,p ))1/2 Un,+ (Sq,p ) −→ Φuniv T

(3.46)

q,p

(l)

Note to this end that the jump times Sq,p of N (l) (q) form a subset of the jump times

(l) Sq,p

of A(l) (q). Finally (3.46) implies (3.25) because of (3.45) and

P(Ω(q, r, n)) → ∞ as n → ∞ for any q > 0 and any r ∈ N. Proof of Theorem 3.6. The proof of Theorem 3.6 is very similar to the proof of Theorem 3.2. Only in Step 1 the used techniques to derive the necessary estimates are somewhat different due to the more complex asynchronous and bivariate structure.

3.1 Central Limit Theorem for Non-Normalized Functionals

97

(l) (r, q) of σ, C as in Step 1. Again we define discretized versions σ ˜ (l) (r, q), C Step 1 in the proof of Theorem 3.2. The only difference here is that we use the stopping times Sq,p , p ∈ N, defined as the common jump times of N (1) (q) and (l) N (2) (q) in the construction instead of the jump times Sq,p , p ∈ N, of N (l) (q). A discretization ρ˜(r, q) of ρ is obtained analogously. Using this notation we then define R(n, q, r) =

√

n





(1)

p:Sq,p ≤T l=1,2

∂l f (ΔNSq,p , ΔN (2) (q)Sq,p )Δ

(l) (l)

in (Sq,p ),n

(l) (r, q) C

and show √ lim lim sup lim sup P(| n(V (f, πn )T − B(f )T ) − R(n, q, r)| > ε) → 0

q→∞ r→∞

(3.47)

n→∞

for all ε > 0. The proof of (3.47) will require some rather technical estimates and is therefore postponed to after the discussion of the general structure of the proof. Step 2. Next we show the X -stable convergence L−s

R(n, q, r) −→ Φbiv T (f, q, r)

(3.48)

for all q > 0 and r ∈ N where Φbiv T (f, q, r) =



˜ × σ

 

∂1 f (ΔN (1) (q)Sq,p , ΔN (2) (q)Sq,p )

Sq,p ≤T (1)

$

+

(r, q)Sq,p −



(1),−

L(Sq,p )USq,p + σ ˜ (1) (r, q)Sq,p



(1),+

R(Sq,p )USq,p

(2)

(˜ σ (1) (r, q)Sq,p − )2 L(1) (Sq,p ) + (˜ σ (1) (r, q)Sq,p )2 R(1) (Sq,p )USq,p



+ ∂2 f (ΔN (1) (q)Sq,p , ΔN (2) (q)Sq,p ) σ ˜ (2) (r, q)Sq,p − ρ˜(r, q)Sq,p − +σ ˜ (2) (r, q)Sq,p ρ˜(r, q)Sq,p





$



(1),−

L(Sq,p )USq,p

(1),+

R(Sq,p )USq,p

+ (˜ σ (2) (r, q)Sq,p − )2 (1 − (˜ ρ(r, q)Sq,p − )2 )L(Sq,p )

+



+ (˜ σ (2) (r, q)Sq,p )2 (1 − (˜ ρ(r, q)Sq,p )2 )R(Sq,p )

1/2

(3)

USq,p (4)

σ ˜ (2) (r, q)Sq,p − )2 L(2) (Sq,p ) + (˜ σ (2) (r, q)Sq,p )2 R(2) (Sq,p )USq,p

 .

To this end note that on the set Ω(n, q, r) where two different jumps of N (q) are further apart than |πn |T and any jump time of N (q) is further away from j2−r than k|πn |T for any j ∈ {1, . . . , T 2r } it holds R(n, q, r)1Ω(n,q,r) =

√

n

 

Sq,p ≤T

∂1 f (ΔN (1) (q)Sq,p , ΔN (2) (q)Sq,p )

98

3 Central Limit Theorems



× σ ˜ (1) (r, q)Sq,p − ΔLn (Sq,p ) W (1) + σ ˜ (1) (r, q)Sq,p ΔRn (Sq,p ) W (1) +σ ˜ (1) (r, q)Sq,p − ΔL(1) (S n

+ ∂2 f (ΔN

(1)

(q)Sq,p , ΔN

q,p )

(2)

W (1) + (˜ σ (1) (r, q)Sq,p ΔR(1) (S



(q)Sq,p ) σ ˜

n

(2)

q,p )

W (1)



(r, q)Sq,p − ρ˜(r, q)Sq,p − ΔLn (Sq,p ) W (1)

+σ ˜ (2) (r, q)Sq,p ρ˜(r, q)Sq,p ΔRn (Sq,p ) W (1) +σ ˜ (2) (r, q)Sq,p − +σ ˜ (2) (r, q)Sq,p

$

$

1 − (˜ ρ(r, q)Sq,p − )2 ΔLn (Sq,p ) W (2)

1 − (˜ ρ(r, q)Sq,p )2 ΔRn (Sq,p ) W (2)

+σ ˜ (2) (r, q)Sq,p − ΔL(2) (S n

q,p

W (2) + σ ˜ (2) (r, q)Sq,p ΔR(2) (S ) n

q,p

W (2) )



1Ω(n,q,r)

where ΔLn (Sq,p ) W (1) denotes the increment of W (1) over the interval corresponding to Ln (Sq,p ) and so forth. (3.48) then follows because Condition 3.5(ii) yields √ (1),− i.e. that nΔLn (Sq,p ) W (1) converges stably in law to L(Sq,p )USq,p . The detailed proof of (3.48) is identical to the proof of (3.25). Step 3. Finally we show biv (|Φbiv lim lim sup P T (f ) − ΦT (f, q, r)| > ε) = 0,

(3.49)

q→∞ r→∞

for all ε > 0. Here (3.49) can be proven similarly as (3.26). Step 4. Combining the results (3.47), (3.48) and (3.49) from Steps 1–3 we obtain the claim (3.21) using Lemma B.6. Proof of (3.47). We start as in the proof of (3.24) and observe that on the set Ω(n, q) where any two different jump times of N (q) in [0, T ] are further apart than 2|πn |T it holds √ | n(V (f, πn )T − B(f )T ) − R(n, q, r)|1Ω(n,q) √    (1) (2) ≤ n f (Δ (1) X (1) , Δ (2) X (2) ) in (Sq,p ),n

Sq,p ≤T

− f (Δ

(1)

− f (Δ

(1)

− f (Δ

(1)

(1)

in (Sq,p ),n (1)

in (Sq,p ),n (1)

in (Sq,p ),n

in (Sq,p ),n

X (1) , Δ

(2) (2)

in (Sq,p ),n

(X (2) − N (2) (q)))

(X (1) − N (1) (q)), Δ

(2)

(X (1) − N (1) (q)), Δ

(2)



(2)

in (Sq,p ),n (2)

in (Sq,p ),n

X (2) ) (X (2) − N (2) (q)))

 

− f (ΔN (1) (q)Sq,p , ΔN (2) (q)Sq,p ) − R(n, q, r)

√   + n (1)

 (2)

Sq,p ≤T j:tj,n ≤T

(1)

(2) f (Δ (1) (1) X (1) , Δj,n (X (2) i (S ),n n

q,p

(3.50)



− N (2) (q)))1i(1) (S (1) ),j  n

q,p

(3.51)

3.1 Central Limit Theorem for Non-Normalized Functionals

+

√   n



(2)

(1)

f (Δi,n (X (1) − N (1) (q)), Δ

(1)

Sq,p ≤T i:ti,n ≤T

√  + n

(1)

in (Sq,p ),n

X (2) )1i,i(2) (S (1) )  n

q,p

(3.52)



(1) f (Δi,n (X (1)

−N

(1)

(2) (q)), Δj,n (X (2)

(1) (2) i,j:ti,n ∨tj,n ≤T

−

 

(2) (2)

99



−N

(2)

(q)))1i,j

 

f (ΔM (1) (q)s , ΔM (2) (q)s )

(3.53)

s≤T

using the shorthand-notation 1i,j = 1{I (1) ∩I (2) =∅} Hence using P(Ω(n, q)) → 1 i,n

j,n

as n → ∞ for any q > 0 it remains to show that (3.50)–(3.53) vanish in the sense of (3.24). Step 1. Using a Taylor expansion (3.50) becomes equal to

√     ∂l f (ΔN (1) (q)Sq,p , ΔN (2) (q)Sq,p )  n Sq,p ≤T

l=1,2

×Δ +

(l) (l)

in (Sq,p ),n

(l) (r, q)) (X (l) − N (l) (q) − C

1  (1) n ∂kl f (ΔN (1) (q)Sq,p + θq,p Δ (l) (X (l) − N (l) (q)), ΔN (2) (q)Sq,p in (Sq,p ),n 2 k,l=1,2

n + θq,p Δ

− f (Δ

(1)

− f (Δ

(1)

− f (Δ

(1)

(2) (2)

in (Sq,p ),n

(1)

in (Sq,p ),n (1)

in (Sq,p ),n (1) in (Sq,p ),n

X

(1)

,Δ



(X (2) − N (2) (q)))

ι=k,l

(2) (2)

Δ

in (Sq,p ),n

(X

(2)

(X (1) − N (1) (q)), Δ

(2)

(X (1) − N (1) (q)), Δ

(2)

(ι) (ι)

in (Sq,p ),n

(X (ι) − N (ι) (q))

− N (2) (q)))

(2)

in (Sq,p ),n (2) in (Sq,p ),n

X (2) )

 

(X (2) − N (2) (q))).

(3.54)

The first sum in (3.54) vanishes in the sense of (3.24) because (3.34) also holds (l) (l) (l) (l) under Condition 3.5 (note that δn,− = Ln (s) + Ln (s), δn,+ = Rn (s) + Rn (s)). Further the second sum vanishes similarly as the term Y3 (n, q, r) in the proof of (3.24) also due to (3.34). For the discussion of the last three sums in (3.54) note that (3.8) implies

 

|f (x, y)| = 

x  x 0

0

 

∂11 f (x

, y)dx dx

 ≤



x  x 0

0 (1,1)

which then together with ∂11 f (x, y) = o(|x|p1 (3.9) yields f (x, y) = o(|x|

(1,1) 2+p1

|y|

(1,1) p2

|∂11 f (x

, y)|dx dx

(3.55) (1,1)

|y|p2

) as |xy| → 0 from

) as |xy| → 0. Similarly we obtain

100

3 Central Limit Theorems (2,2)

(2,2)

f (x, y) = o(|x|p1 |y|2+p2 ) as |xy| → 0. Hence the sums over the expressions in the last three lines of (3.54) vanish also by (3.34), compare the discussion of Y2 (n, q, r) in the proof of (3.24). (l) Step 2. Denote by Ω(N, q, m, n) the set where |Δi,n X (l) | ≤ N and (l)

|Δi,n (X (l) − N (l) (q))| ≤ 1/m (l)

for any ti,n ≤ T , l = 1, 2. It holds P(Ω(N, q, m, n)) → 1 as N, n → ∞ for any m, q > 0. On the set Ω(N, q, m, n) the sums (3.51) and (3.52) are using (1,1)

(1,1)

(2,2)

f (x, y) = o(|x|2+p1 |y|p2 |xy| → 0 bounded by √  KN,m n



) as |xy| → 0 and f (x, y) = o(|x|p1



l=1,2 S (l) ≤T j:t(3−l) ≤T q,p j,n

|Δ

(2,2)

|y|2+p2

) as

(3−l,3−l)

(1) (1) (1) in (Sq,p ),n

X (1) |pl

(3−l,3−l)

(3−l)

× |Δj,n (X (3−l) − N (3−l) (q))|2+p3−l

1i(1) (S (1) ),j . n

q,p

By similar arguments as used in the derivation of (3.34) we can deduce using inequalities 1.8–1.10

E



(3−l)

j:tj,n

(3−l,3−l)

(3−l)

|Δj,n (X (3−l) − N (3−l) (q))|2+p3−l

  1i(1) (S (1) ),j σ(G0 (q), S) n

q,p

≤T (3−l,3−l)

≤ K(Kq (|πn |T )1+p3−l

(3−l,3−l)

+ (|πn |T )p3−l

/2

+ eq )3|πn |T .

Hence the sums (3.51) and (3.52) vanish as n → ∞ for any q > 0 because N (l) (q), l = 1, 2, almost surely has only finitely many jumps in [0, T ] and because we have √ n|πn |T = oP (1) by Condition 3.5(i). Step 3. Denote by (l,3−l)



Δ(i,j),n M (l) (q) = M (l) (q)t(l) ∧t(3−l) − M (l) (q)t(l) i,n

(l\3−l)

(3−l) i−1,n ∨tj−1,n

j,n

(l)



1i,j ,

(l,3−l)

Δ(i,j),n M (l) (q) = Δi,n M (l) (q) − Δ(i,j),n M (l) (q) (l)

(3−l)

(l)

(3−l)

the increments of M (l) (q) over the sets Ii,n ∩ Ij,n respectively Ii,n \ Ij,n . Then using the above notation and a Taylor expansion the sum (3.53) is equal to √  n

 (1)

(1,2)

(1,2)

f (Δ(i,j),n M (1) (q), Δ(i,j),n M (2) (q))1i,j

(2)

i,j:ti,n ∨tj,n ≤T

−

 s≤T

f (ΔM (1) (q)s , ΔM (2) (q)s )

 (3.56)

3.1 Central Limit Theorem for Non-Normalized Functionals

+

√



n



(1,2)

101

(1,2)

∂l f (Δ(i,j),n M (1) (q), Δ(i,j),n M (2) (q))

(1) (2) i,j:ti,n ∨tj,n ≤T l=1,2

(l)

+

√

n







(1,2) M (1) (q) 1 ,i2 ),n

2

(1) + θin1 ,i2 [Δi1 ,n (B (1)

(1\2) M (1) (q)], 1 ,i2 ),n

+ C (1) ) + Δ(i

(1,2) (2) M (2) (q) + θin1 ,i2 [Δi2 ,n (B (2) 1 ,i2 ),n

Δ(i

(3.57)

∂kl f Δ(i

k,l=1,2 i ,i :t(1) ∨t(2) ≤T 1 2 i ,n i ,n 1

(l\3−l)

× (Δi,n (B (l) + C (l) ) + Δ(i,j),n M (l) (q))1i,j

+ C (2) )

 (k) (2\1) M (2) (q)] (Δik ,n (B (k) 2 ,i1 ),n

+ Δ(i

(l)

(k\3−k) M (k) (q)) k ,i3−k ),n

+ C (k) ) + Δ(i

(l\3−l) M (l) (q))1i1 ,i2 . l ,i3−l ),n

× (Δil ,n (B (l) + C (l) ) + Δ(i

(3.58)

Then the S-conditional expectation of the sum corresponding to l = 1 in (3.57) (l) is on the set Ω(n, m, q) where it holds Δi,n M (l) (q) ≤ 1/m, for all i ∈ N with (l)

ti,n ≤ T , l = 1, 2, using iterated expectations, the Cauchy-Schwarz inequality as in the discussion of Y5 (n, q) in the proof of (3.24), Lemma 1.4 and Lemma 3.9 bounded by  (1,1) (1,1)

√ (1,2) (1,2) n E Km |Δ(i,j),n M (1) (q)|1+p1 |Δ(i,j),n M (2) (q)|p2 | (1)

(2)

i,j:ti,n ∨tj,n ≤T (l)

 

(l\3−l)

× (Δi,n (B (l) + C (l) ) + Δ(i,j),n M (l) (q))|S 1i,j ≤ Km e q G n 2,1 (T ). Analogously the S-conditional expectation of the sum in (3.57) corresponding to l = 2 can be bounded by Km eq Gn 1,2 (T ). Further on the set Ω(n, m, q) the expression (3.58) is similarly as Y6 (n, q) in the proof of (3.24) bounded by Km



k,l=1,2

Gn(k,l) p1

(k,l)

+(2−k)+(2−l),p2

+(k−1)+(l−1)

(T ).

Hence (3.57) and (3.58) vanish as n → ∞ for any q > 0 because of lim P(Ω(n, m, q)) = 0

n→∞

for any q, m > 0, Km → 0 as m → 0 and because of Condition 3.5(i) and n n Gn p,3−p (T ) ≤ G3,0 (T ) + G0,3 (T ), p ∈ [0, 3],

(3.59)

where the inequality follows from ap b3−p ≤ a3 + b3 for any a, b ≥ 0 which can be concluded from Muirhead’s inequality; compare Theorem 45 of [19]. Finally (3.56) can be discussed similarly as the term Y4 (n, q) in the proof of (3.24).

102

3 Central Limit Theorems

3.2 Central Limit Theorem for Normalized Functionals In this section we will discuss how central limit theorems for the normalized (l) functionals V (p, g, πn )T and V (p, f, πn )T introduced in Section 2.2 can be obtained. To illustrate usable methods and arising challenges we are going to discuss the asymptotics of

 √  n V (p, f, πn )T −



T 0

mcs (f )dGp (s)

(3.60)

under synchronous observations. Hence we will outline how to derive a central limit theorem supporting the law of large numbers from Theorem 2.17. In the end of this section we then briefly discuss why it would be much more difficult to obtain a similar result in the setting of asynchronous observations. The additional assumptions which are necessary to derive central limit theorems (l)

for V (p, g, πn )T and V (p, f, πn )T in the case of synchronous observations are summarized in the following condition. From now on we use the notation from (2.1). Condition 3.10. Suppose Condition 1.3 holds, the observation scheme is exogenous and the process X is continuous. Further, (i) the volatility process is a continuous 2 × 2-dimensional semimartingale, i.e.

 σt = σ0 +

t 0

˜bs ds +

  l=1,2 0

t

(l)

(l)

σ ˜s dWs +



t 0

τ˜s dVs

(3.61)

(l)

where ˜bs , σ ˜s , τ˜s are adapted 2 × 2-dimensional processes with c` adl` ag paths and V is a 2×2-dimensional standard Brownian motion which is independent t (l) (l) of W . The integrals 0 σ ˜s dWs are to be understood in such a way that (l)

each component of σ ˜s

(l)

is integrated with respect to Ws .

(ii) The observation scheme is given by t0,n = 0, n ∈ N, and ti,n = ti−1,n +

Ei,n , i, n ∈ N, n

for i.i.d. non-negative random variables Ei,n , i, n ∈ N, whose law fulfils k E[E1,1 ] < ∞ for all k ≥ 0.

3.2 Central Limit Theorem for Normalized Functionals

103

In the following we will sketch how based on Condition 3.10 a central limit theorem can be obtained for V (p, f, πn )T where the function f : R2 → R is continuously differentiable and positively homogeneous of degree p ≥ 1 in the (1),n (2),n (t) = Gp (t) in this setting of synchronous observations. Set Gn p (t) = Gp setting and denote by Gp (t) the (deterministic) limit of Gn p (t) which always exists under Condition 3.10(ii) by Lemma 5.3. First we decompose (3.60) as follows

 √  n V (p, f, πn )T − √

= nn + +



p/2−1

√ √ √

mcs (f )dGp (s)

0



f (Δi,n X) − f (σti−1,n Δi,n W )

i:ti,n ≤T



nnp/2−1



n

i:ti,n ≤T

i:ti,n ≤T

+



T



n

i:ti,n ≤T





(3.62)

f (σti−1,n Δi,n W ) − mcti−1,n (f )|Ii,n |p/2



(3.63)





mcti−1,n (f ) np/2−1 |Ii,n |p/2 − (Gp (ti,n ) − Gp (ti−1,n )) (3.64)



ti,n ti−1,n

(mcti−1,n (f ) − mcs (f ))dGp (s) + oP (1)

(3.65)

where cs = σs σs∗ . Here, (3.62) and (3.65) describe errors which are due to a discretization of σ. Further, the term  (3.63) originates from the error in the convergence of the standardized sum n−1 i np/2 |Ii,n |p/2 f (σti−1,n Δi,n W |Ii,n |−1/2 ) to corresponding moments of a normal distribution. The term (3.64) represents the error in the convergence of the observation scheme. The oP (1)-term is due to the integral √



T

n tin (T ),n

√ mcs (f )dGp (s) = OP ( n|Iin (T ),n |) = OP (n−1/2 ).

To derive stable convergence of (3.60) we want to use Proposition C.4 or rather a generalized version of it. Hence we have to check (C.1)–(C.5) for ζin =

 √  p/2−1 n n f (Δi,n X) −

ti,n

 mcs (f )dGp (s) .

ti−1,n

Instead of checking the conditions (C.1)–(C.5) for ζin we will check them for the corresponding increments of (3.62)–(3.65) separately. First we consider (3.63). Setting



αin = n(p+1)/2−1 f (σti−1,n Δi,n W ) − mcti−1,n (f )|Ii,n |p/2



104

3 Central Limit Theorems

we obtain E[αin |Fti−1,n , S] = 0 for all i, n ∈ N which yields



E[αin |Fti−1,n , S] = 0,

i:ti,n ≤T





E[(αin )2 |Fti−1,n , S] = np−1

i:ti,n ≤T P

−→

i:ti,n ≤T



T 0

(mcs (f 2 ) − (mcs (f ))2 )dG2p (s).

(l)

(mcti−1,n (f 2 ) − (mcti−1,n (f ))2 )|Ii,n |p



Denote mΣ (f, 1) = E g(Σ1/2 Z)Z (l) where Z = (Z (1) , . . . , Z (d) ) ∼ N (0, Id ) for covariance matrizes Σ ∈ Rd×d . Then we derive for l = 1, 2



E[αin Δi,n W (l) |Fti−1,n , S]

i:ti,n ≤T



= n(p+1)/2−1 P

i:ti,n ≤T



−→

T 0

(l)

(mcti−1,n (f, 1) − mcti−1,n (f )mI1 (id))|Ii,n |(p+1)/2

(l)

mcs (f, 1)dGp+1 (s)

where we used mI1 (id) = 0. Next we look at (3.64). To discuss this term we need Condition 3.10(ii) as here we have to work with the asymptotics of the observation scheme. Note that Condition 3.10(ii) and Lemma 5.3 yield k/2

Gk (t) =

E[E1,1 ] t, ∀t ≥ 0, ∀k ≥ 0. E[E1,1 ]

(3.66)

Hence we have √



n

i:ti,n ≤T

=

√

   E mcti−1,n (f ) np/2−1 |Ii,n |p/2 − (Gp (ti,n ) − Gp (ti−1,n )) Fti−1,n



n

i:ti−1,n ≤T

=n

−1/2



i:ti−1,n ≤T

=0

mcti−1,n (f )E n−1 Ei,n − E[E1,1 ]/E[E1,1 ]|Ii,n | p/2

p/2

p/2

p/2

mcti−1,n (f )E Ei,n − E[Ei,n ]/E[Ei,n ]Ei,n





3.2 Central Limit Theorem for Normalized Functionals and  i:ti,n ≤T

 √  2  E ( nmcti−1,n (f ))2 np/2−1 |Ii,n |p/2 − (Gp (ti,n ) − Gp (ti−1,n )) |Fti−1,n



=

i:ti,n ≤T



=

i:ti,n ≤T

  p/2 n(mcti−1,n (f ))2 E (np/2−1 |Ii,n |p/2 − E[Ei,n ]/E[Ei,n ]|Ii,n |)2

n (mcti−1,n (f ))2 (Gn 2p (ti,n ) − G2p (ti−1,n )) p/2

E[Ei,n ]

−2 P



−→

105

T 0

E[Ei,n ]

E[Ei,n ]2 p/2

n (Gn p+1 (ti,n ) − Gp+1 (ti−1,n )) +

E[Ei,n

]2

n (Gn 4 (ti,n ) − G4 (ti−1,n ))



(Gp )2  Gp (mcs (f )) d G2p − 2 Gp+1 + G4 (s) G2 (G2 )2 

2

where we used (3.66) repeatedly. Here, the fraction Gp (t)/G2 (t) does not depend on t. Finally we consider the error terms (3.62) and (3.65) which are arising from the discretization of σ. In particular we need to quantify the approximation error due to this discretization and therefore need the structural assumption (3.61) on the form of σ. Further we will use the differentiability of f for the discussion of these terms. Using a Taylor expansion we get √



n



i:ti,n ≤T

≈

 i:ti,n ≤T

ti,n ti−1,n



(mcti−1,n (f ) − mcs (f ))dGp (s)

ti,n ti−1,n

√ (∇mcti−1,n (f ))∗ n(κ(σti−1,n ) − κ(σs ))dGp (s)

where κ(v) = (v1,1 , v1,2 , v2,1 , v2,2 )∗ for v =



v1,1 v2,1

v1,2 v2,2

∗

∈ R2×2

denotes the operation that tranforms a 2 × 2-matrix into a 4 × 1-vector and ∇mx∗ x (f ) =

 ∂mx∗ x (f ) ∂mx∗ x (f ) ∂mx∗ x (f ) ∂mx∗ x (f )  ∂x1,1

,

∂x2,1

,

∂x1,2

,

∂x2,2

denotes the derivative of mΣ (f ) with respect to the entries of the root Σ1/2 of the covariance matrix. Note that f is for p ≥ 1 regular enough, compare Lemma 2.28, such that e.g. ∂mx∗ x (f ) ∂ ∂ = E[f (xZ)] = E[ f (xZ)], Z ∼ N (0, I2 ), ∂x1,1 ∂x1,1 ∂x1,1

106

3 Central Limit Theorems

holds and hence the existence of ∇mx∗ x (f ) is assured becaue f is continuously differentiable. Further, we infer

E[Δi,n σ|Fti−1,n , S] = OP (|Ii,n |), E[Δi,n σ2 |Fti−1,n , S] = OP (|Ii,n |) from (3.61), compare Lemma 1.4, which yields





P

E[βin |Fti−1,n , S] −→ 0,

i:ti,n ≤T

P

E[(βin )2 |Fti−1,n , S] −→ 0,

i:ti,n ≤T

∗ ti,n

√ where βin = (∇mcti−1,n (f )) t n(κ(σti−1,n ) − κ(σs ))dGp (s). Hence (3.65) i−1,n is asymptotically negligible by Lemma C.3. t Finally we consider (3.62). Note that we can write Xt = 0 bs ds + Ct for some adapted process bs because X is continuous by Condition 3.10. By a Taylor expansion we then obtain √  p/2−1 n n

 i:ti,n ≤T

≈n

(∇f (σti−1,n Δi,n W ))∗

i:ti,n ≤T

×





ti,n ti−1,n

=n





ti,n



ti,n

(σs − σti−1,n )dWs

bs ds +



ti−1,n



(p+1)/2−1



(∇f (σti−1,n Δi,n W ))∗

ti−1,n

≈n

σs dWs − σti−1,n Δi,n W

ti−1,n



i:ti,n ≤T

×

ti,n

bs ds +

(p+1)/2−1

f (σti−1,n Δi,n W )

i:ti,n ≤T



(p+1)/2−1





f (Δi,n X) − np/2−1

(∇f (σti−1,n Δi,n W ))∗

i:ti,n ≤T



× bti−1,n |Ii,n | +

 

ti,n ti−1,n

(l)

(l)

(l)

σ ˜ti−1,n (Ws − Wti−1,n )

l=1,2



+ τ˜ti−1,n (Vs − Vti−1,n ) dWs



where we used (3.61) for the last approximation. Set



γin = n(p+1)/2−1 (∇f (σti−1,n Δi,n W ))∗ bti−1,n |Ii,n |



ti,n

+ ti−1,n

  l=1,2

(l)

(l)

(l)





σ ˜ti−1,n (Ws − Wti−1,n ) + τ˜ti−1,n (Vs − Vti−1,n ) dWs .

3.2 Central Limit Theorem for Normalized Functionals

107

Using the fact that if f is positively homogeneous of degree p ≥ 1 and differentiable then ∇f is positively homogeneous of degree p − 1, compare Theorem 20.1 in [44], we then derive  E[γin |Fti−1,n , S] i:ti,n ≤T

 |Ii,n |(p+1)/2 E[(∇f (σti−1,n Δi,n W/|Ii,n |1/2 ))∗ |Fti−1,n , S]



= n(p+1)/2−1

i:ti,n ≤T

× bti−1,n + E[(∇f (σti−1,n Δi,n W/|Ii,n |)1/2 ))∗  1 ti,n    (l) × σ ˜ti−1,n Ws(l) dWs |Fti−1,n , S] |Ii,n | ti−1,n l=1,2

 = n(p+1)/2−1 |Ii,n |(p+1)/2 (mcti−1,n (∇f ))∗ bti−1,n i:ti,n ≤T

+ E[(∇f (σti−1,n V1 ))∗ P

−→



T 0

(l)





1 |Ii,n |

l=1,2

(mcs (∇f ))∗ bs dGp+1 (s) + (l)



(l)

σ ˜ti−1,n



T 0

c m

1 0

  Vs(l) dVs |Fti−1,n ]

(1) (2) cs ,˜ cs s ,˜

(∇f )dGp+1 (s)

(l)

where c˜s = σ ˜s (˜ σs )∗ and V = (V (1) , V (2) )∗ denotes a two-dimensional standard Brownian motion independent of F and m c





(1) (2) cs ,˜ cs s ,˜

 (V (1) )2 − 1   1 V (1) dV (2)    (2) 1 s 0 s ˜s 1 1 (2) (1) + σ F . (2) 2 2 V dVs (V1 )2 − 1 0 s

(1) 1

(∇f ) = E (∇f (σs V1 ))∗ σ ˜s

Summarizing the investigation of the conditions (C.1)–(C.5) we observe that for (C.1) only (3.62) yields a non-zero contribution. Further, for condition (C.2) we find that the terms (3.63) and (3.64) contribute to the limit while for condition (C.3) only (3.63) yields a non-zero contribution. The term (3.65) does not contribute to the limit at all. Combining all results from above according to Proposition C.4 we obtain the stable convergence

 √  n V (p, f, πn )T − L−s

−→



T 0

+





T 0

mcs (f )dGp (s)

(mcs (∇f ))∗ bs + m c

  l=1,2 0

T

$

(l)



(1) (2) cs ,˜ cs s ,˜

(∇f ) dGp+1 (s) (l)

mcs (f, 1)G p+1 (s)dWs +



T 0

√

s , (3.67) ws dW

108

3 Central Limit Theorems



ws = mcs (f 2 )G 2p (s) −

l=1,2

(l)

mcs (f, 1)G p+1 (s)



+ (mcs (f ))2 −

Gp (Gp )2 

Gp+1 + G4 (s) G2 (G2 )2

 denotes a standard Brownian motion which is independent of F . By where W Lemma 5.3 it is clear that all functions Gp , p ≥ 0, are differentiable such that the expressions above are well defined. We did not formally check conditions (C.4) and (C.5) but they should follow similarly as in the setting of equidistant deterministic observations which is discussed in [40]. Further it should be checked that no covariations of the terms (3.62)-(3.65) enter the limit. The structure of the limit (3.67) is apart from the terms originating from the irregular observation scheme similar to Theorem 3.3 in [40]. However, due to the bivariate setting for example the term m  c ,˜c(1) ,˜c(2) (∇f ) has a much more s s s complicated structure compared to the corresponding term in the univariate setting. p1 p2 If f is an even function e.g. f (x1 , x2 ) = |x1 | |x2 | we have that ∇f is odd and (l) mcs (f, 1), mcs (∇f ), m  c ,˜c(1) ,˜c(2) (∇f ) all vanish in that case. Hence we obtain s

√  n V (p, f, πn )T − L−s

−→



T 0

$



s

s



T 0

mcs (f )dGp (s)





s mcs (f 2 )G 2p (s) + (mcs (f ))2 − Gp Gp+1 /G2 + Gp 2 G4 /G2 2 (s)dW

for even functions f : R2 → R. This corresponds to Theorem 14.3.2 in [30] as there our functions Gp (t) are equal to the integrals



t 0

(p+1)/2−1

m p/2 θs

ds

using their notation. In the case of asynchronous observation times we would assume that both the observation times of X (1) and X (2) are of the form as in Condition 3.10(ii). Next, also in the asynchronous setting it would be reasonable to start with a decomposition similar to (3.62)–(3.65). However, it is not straightforward anymore how the volatility process should be discretized. As seen above this discretization error contributes to the limit and hence the choice of the discretization is very important. Additionally, the summands in V (p, f, πn )T in the asynchronous setting do not contain increments over disjoint intervals which makes a direct application of Proposition C.4 infeasible. Further, especially in the situation of Theorem 2.22 where already the limit of V (p, f, πn )T has a structure which is much more complicated than in the synchronous setting it is to be expected

3.2 Central Limit Theorem for Normalized Functionals

109

that also the asymptotic variances will have a form which is much more involved. However, putting the technical difficulties sketched above aside it is plausible by the exogeneity of the observation scheme and its specific form assumed in Condition 3.10(ii) that a central limit theorem of some kind should indeed hold. Remark 3.11. In this remark we discuss the assumptions made in Condition 3.10: The form of the volatility (3.61) was necessary to estimate an error due to the discretization of σ and is also required in the setting of equidistant deterministic observation times; compare (3.18) in [40]. As we here consider a two-dimensional process X we have to consider a matrix-valued process σ. Note that in Section 14.3.1 of [30] a central limit theorem is only derived for one-dimensional processes stating “the result for k ≥ 2 being much more difficult to prove”. Maybe we could also allow the volatility process σ to be a discontinuous Itˆ o semimartingale as in Assumption (K) in Section 14.3.1 of [30]. Condition 3.10(ii) guarantees the convergence of Gn p (t) for all p ≥ 0; compare Lemma 5.3. A weaker assumption which would yield a central limit theorem for Gn  p (t), t ≥ 0, might also be sufficient. Notes. The methods used in this section follow the arguments in Section 3.1 of [40]. Further, we compared our computations to those leading up to Theorem 14.3.2 in [30]. Like this, we made sure that the structure of the limit (3.67) at least coincides in the more simple cases with the results from Theorem 3.3 in [40] and Theorem 14.3.2 in [30]. Already in the synchronous setting, we observed that the asymptotic variance in (3.67) has a rather complicated form and should therefore be difficult to estimate. To this end note that the Observed Asymptotic Variance approach discussed in [39] allows to construct asymptotic confidence intervals also without knowing the explicit form of the limit. There, under the assumption that a central limit theorem of some form holds, the asymptotic variance is estimated via a bootstrapping approach directly from the available observations. This has the remarkable advantage that no knowledge of the form of the central limit theorem is required. Hence, if we believe that a central limit theorem also holds in the asynchronous setting corresponding to the situations in Theorems 2.21 or 2.22 we could use their method to estimate the asymptotic variance. However, to formally apply this procedure we have to prove that a central limit theorem of some kind holds which might not be much easier than computing its specific form.

4 Estimating Asymptotic Laws In order to make use of the central limit theorems 3.2 and 3.6 the law of the limiting variables univ,(l)

ΦT

(g) =



(l)

(l)

(l)

(l)

(l)

(l)

(l)

g (ΔXs )(σs− (δ− (s))1/2 U− (s) + σs (δ+ (s))1/2 U+ (s))

s≤T

and Φbiv T (f ) or at least a suitable estimate of their law has to be known to the (l)

statistician. Denote by Sp the jump times of X (l) and by Sp the common jump times of X (1) and X (2) . If we look through the components making up the limits univ,(l) (l) ΦT (g) and Φbiv respectively X, T (f ), we find that the jump heights of X (l) (1) 2) the values and left limits of σ respectively σ , σ , ρ at the jump times and (l) the laws Γuniv,(l) (Sp , dy), Γbiv (Sp , dy) are necessary to describe the asymptotic law but in general unknown to the statistician. To estimate the jump heights we may use the threshold technique introduced in Section 2.3 to identify unusually“ large increments as jumps. Further we ” will devote ourselves to the estimation of σ (l) , l = 1, 2, and ρ at specific times 1 in Chapter 6 . Hence it remains the task to find suitable estimates for the laws (l) Γuniv,(l) (Sp , dy) and Γbiv (Sp , dy). First note that without making additional assumptions on the observation scheme it is impossible to complete this task. This is due to the fact that the laws Γuniv,(l) (s, dy), Γbiv (s, dy) are dependent on the time variable s and the only information on the asymptotic law Γuniv,(l) (s, dy), Γbiv (s, dy) which we can extract from the observed data are the realizations (l) (l) (nδn,− (s), nδn,+ (s)) respectively Znbiv (s) whose law converges to Γuniv,(l) (s, dy), Γbiv (s, dy). Here we are given only a single realization according to the approximate asymptotic law no matter how large n is and hence there is no hope to find a consistent estimator for the asymptotic laws. To circumvent the issue sketched above we impose an asymptotic local homogeneity assumption on the observation scheme πn . In particular we require that the laws Γuniv,(l) (s , dy), Γbiv (s , dy) are close to Γuniv,(l) (s, dy), Γbiv (s, dy) if s is close to s. In such a situation we can estimate the laws Γuniv,(l) (s, dy), Γbiv (s, dy) (l) (l) from the realizations (nδn,− (s ), nδn,+ (s )) respectively Znbiv (s ) for multiple s

1 Although

these estimators are already needed in this chapter, I decided to discuss the (l) estimation of σs and ρs in the applications part because these quantities are also of individual interest in certain applications.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 O. Martin, High-Frequency Statistics with Asynchronous and Irregular Data, Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics, https://doi.org/10.1007/978-3-658-28418-3_4

112

4 Estimating Asymptotic Laws

which are close to s. We are then able to consistently bootstrap the asymptotic laws using this procedure if we pick s from a shrinking neighbourhood [s − εn , s + εn ] of s such that εn → 0 as n → ∞ but also in such a way that the neighbourhood is (l) (l) (l) (l)

large enough such that (nδn,− (s ), nδn,+ (s n )), (nδn,− (s

n ), nδn,+ (sn )) respectively



Znbiv (s n ), Znbiv (s

n ) become asymptotically independent for ”typical” sn , sn chosen from [s − εn , s + εn ].

As in Chapter 3 we will use the discussion in this chapter concerning the (l) estimation of the laws Γuniv,(l) (Sp , dy), Γbiv (Sp , dy) to introduce and analyse general methods which will later be used in the applications in Chapters 7–9 as well.

4.1 The Results To formalize the bootstrap idea sketched above let (Kn )n∈N and (Mn )n∈N denote increasing sequences of natural numbers tending to infinity as n → ∞. As in Section 3.1 we first discuss the univariate case because the notation and the results are structurally simpler in that case. We then define



(l) (l) (l) δˆn,m,− (s) = nκn,m (s)I (l)

(l) in (s)+Vn,m (s),n

 ,



(l) (l) (l) δˆn,m,+ (s) = n(1 − κn,m (s))I (l)

(l) in (s)+Vn,m (s),n

 

(4.1)

(l)

for m = 1, . . . , Mn where κn,m (s) denotes a U[0, 1]-distributed random variable (l)

and Vn,m (s) is distributed according to

 (l) (l) P(Vn,m (s) = k|S) = I (l)

in (s)+k,n

 

Kn 

 (l) I (l)

k =−Kn

in (s)+k ,n

−1  ,

(4.2)

(l)

k ∈ {−Kn , . . . , Kn }. See Figure 4.1 for an illustration. The variables κn,m (s) and

(l) ) on which  F, P Vn,m (s) are defined on the same extended probability space (Ω, univ,(l)

ΦT (g) is defined. They are apart from the property (4.2) independent of F and they are conditionally on F independent of each other for different m and/or (l) (l) different s. Note that by the above construction (δˆn,m,− (s), δˆn,m,+ (s)) is equal to



(l)

nδn,− (t

(l) (l)

(l)

(l)

in (s)+Vn,m (s)−1,n (l)



nδn,+ t

+ κn,m (s)|I

(l) (l)

(l) (l)

(l)

in (s)+Vn,m (s),n (l)

(l)

in (s)+Vn,m (s)−1,n

|),

+ κn,m (s)|I

(l) (l)

(l)

in (s)+Vn,m (s),n

|)



4.1 The Results

113

(l) δˆn,m,− (s)

(l)

I (l)

(l) δˆn,m,+ (s)

in (s),n

X (l) (l)

(l)

t (l)

in (s)−2−1,n

(l)

+ κn,m (s)|I (l)

in (s)−2,n

s

|

(l) (l) (l) Figure 4.1: Realization of (δˆn,m,− (s), δˆn,m,+ (s)) for Vn,m (s) = −2.

where the argument t

(l)



(l)

(l)

(l)

in (s)+Vn,m (s),n

+ κn,m (s)I

(l) (l)

(l)

in (s)+Vn,m (s),n

  denotes a ran-

dom variable which is conditionally on S uniformly distributed on the interval [t

(l) (l)

in (s)−Kn −1,n

,t



(l) (l)

in (s)+Kn ,n

.

(4.3)

Remark 4.1. The interval (4.3) corresponds to the interval [s − εn , s + εn ] used in the introductory sketch and the above construction corresponds to picking s

precisely Mn times uniformly from this interval. To obtain this property we had (l) to weight the probabilities by which Vn,m (s) chooses from the 2Kn + 1 intervals I

(l)

(l)

in (s)+k,n

, k ∈ {−Kn , . . . , Kn }, surrounding s by the corresponding interval

length. This construction relates to Condition 3.1(ii) where the outer integral in (3.17) over the time variables can be interpreted as an expectation with respect to univ,(l) uniform random variables; compare Example 3.3. Also in the limit ΦT (g) the (l)

(l)

(l)

variable (δ− (·), δ+ (·)) only occurs evaluated at jump times Sp of N (l) (q) and the jump times can under certain conditions be assumed to be uniformly distributed; compare the proof of (3.25).  Using the variables defined in (4.1) and i.i.d. standard normal distributed random (l),− (l),+ variables Un,i,m , Un,i,m defined on the extended probability space and independent of all previously introduced random variables we define by

% Φ T,n,m (g) = univ,(l)

 (l)



(l)

g (Δi,n X (l) )1{|Δ(l) X (l) |>β|I (l) | } i,n

i:ti,n ≤T (l)

(l)

(l)

(l)

(l),−

i,n

(l)

(l)

(l)

(l)

(l),+ 

× σ ˜n (ti,n , −)(δˆn,m− (ti,n ))1/2 Un,i,m + σ ˜n (ti,n , +)(δˆn,m,+ (ti,n ))1/2 Un,i,m , m = 1, . . . , Mn , random variables whose F -conditional distribution approximates univ,(l) the X -conditional distribution of ΦT (g). Here, we set β > 0,  ∈ (0, 1/2), compare Section 2.3, to separate large increments likely dominated by the jump part of X (l) from small increments likely dominated by the continuous martingale

114

4 Estimating Asymptotic Laws (l)

(l)

(l)

(l)

part of X (l) . σ ˜n (ti,n , −) and σ ˜n (ti,n , +) denote estimators for σ

(l) (l)

ti,n −

, σ

(l) (l)

ti,n

which will be formally introduced in Chapter 6. % univ,(l) (g), m = 1, . . . , Mn , are F -conditionally By construction the variables Φ T,n,m independent and identically distributed. Hence if we let Mn → ∞ for fixed n their % univ,(l) (g). empirical distribution converges to the F -conditional distribution of Φ T,n,1

% On the other hand we defined Φ T,n,1

univ,(l)

(g) such that its F -conditional distribution univ,(l)

univ,(l)

approximates the X -conditional distribution of ΦT (g). Note that ΦT (g) depends on F only through X . Hence we can use the empirical distribution on % univ,(l) (g), m = 1, . . . , Mn , to estimate the X -conditional law of the limiting the Φ T,n,m univ,(l)

variable ΦT (g) which we were interested in from the beginning. For the construction of asymptotic confidence intervals not the whole distribution univ,(l) of ΦT (g) is necessary but only specific quantiles of the distribution. To this end we define

% Q T,n

univ,(l)

%α ({Φ % (g, α) = Q T,n,m (g)|m = 1, . . . , Mn }) univ,(l)

% to be the αMn -largest element of the set {Φ T,n,m (g)|m = 1, . . . , Mn }. Hence univ,(l)

% by construction Q T,n

univ,(l)

tribution of the

(g, α) corresponds to the α-quantile of the empirical dis-

% univ,(l) (g), Φ T,n,m

m = 1, . . . , Mn . Then following the above considera-

%univ,(l) (g, α) converges under appropriate conditions to the X -conditional tions Q T,n univ,(l)

univ,(l)

α-quantile QT (g, α) of ΦT (g) which is defined as the (under certain conditions unique) X -measurable [−∞, ∞]-valued random variable fulfilling

    Φuniv,(l) (g) ≤ Quniv,(l) (g, α)X (ω) = α, P T T

ω ∈ Ω.

(4.4)

In the following condition, we state a precise assumption which yields the local homogeneity that we had to require in the introductory motivation such that our bootstrap procedure is able to consistently estimate the distribution of the limit in Theorem 3.2. Further, some additional structural assumptions have to be made. Condition 4.2. Suppose Condition 3.1 is fulfilled, the set {s ∈ [0, T ] : g (ΔXs ) = (l) 0} is almost surely not empty, the set {s ∈ [0, T ] : σs = 0} is almost surely a T univ,(l) Lebesgue null set and 0 Γ (s, {0})ds = 0. Further, let the sequence (bn )n∈N P

used for the estimators σ ˜ (·, −), σ ˜ (·, +) fulfil bn → 0, |πn |T /bn −→ 0 and suppose that (Kn )n∈N and (Mn )n∈N are sequences of natural numbers converging to infinity with Kn /n → 0. Additionally:

4.1 The Results

115

(i) It holds

 (l) (l)   P((δˆn,1,− (sp ), δˆn,1,+ (sp )) ≤ xp , p = 1, . . . , P |S) P −

P 

  ((δ (l) (sp ), δ (l) (sp )) ≤ xp ) > ε → 0 P − +

p=1

as n → ∞, for all ε > 0 and any P ∈ N, xp ∈ R2 and sp ∈ (0, T ), p = 1, . . . , P , with sp = sp for p = p . (ii) The volatility process σ (l) is itself an Itˆ o semimartingale i.e. a process of the form (1.1). Assuming that the set {s ∈ [0, T ] : g (ΔXs ) = 0} is almost surely not empty, (l) that the set {s ∈ [0, T ] : σs = 0} is almost surely a Lebesgue null set and T univ,(l) that 0 Γ (s, {0})ds = 0 holds is necessary to guarantee that the limit univ,(l)

univ,(l)

(g) is almost surely non-singular because on the set where ΦT (g) ΦT univ,(l) (g, α) is not well-defined in the is singular the X -conditional quantile Q sense of (4.4). Condition 4.2(i) contains the local homogeneity assumption and (l) (l) yields that the S-conditional distribution of (δˆn,1,− (s), δˆn,1,+ (s)) converges to Γuniv,(l) (s, dy). The sequence (bn )n∈N and part (ii) of Condition 4.2 are needed (l) (l) for the consistency of the spot volatility estimators σ ˜ (l) (ti,n , −), σ ˜ (l) (ti,n , +). Under Condition 4.2 we obtain

% Q T,n

univ,(l)

P

univ,(l)

(g, α) −→ QT

(g, α)

(4.5)

which is used to obtain the following result. Theorem 4.3. Suppose Condition 4.2 is fulfilled and g fulfils the assumptions made in Theorem 3.2. Then we have for α ∈ [0, 1]

 lim P

n→∞

√

% n(V (l) (g, πn )T − B (l) (g)T ) ≤ Q T,n

univ,(l)

 

(g, α)F = α

(4.6)

for any F ∈ X with P(F ) > 0. Here, we are able to consider X -conditional quantiles and to obtain the convergence in (4.6) conditional on sets F ∈ X because we are working with the X -stable convergence shown in Theorem 3.2. Using (4.6) we can construct an asymptotic symmetric confidence interval with level α for B (l) (g)T via

% V (l) (g, πn )T − n−1/2 Q T,n

univ,(l)

% (g, (1 + α)/2), V (l) (g, πn )T + n−1/2 Q T,n

univ,(l)

(g, (1 + α)/2)



116

4 Estimating Asymptotic Laws univ,(l)

based on the fact that the F -conditional distribution of ΦT (g) is symmetrical. Next we consider the bivariate situation where we need to estimate the law of Φbiv T (f ). Compared to the univariate situation the theory here is a little more complicated because now we have to locate s in the observation scheme both of X (1) and of X (2) and also because the general structure of the limit in Theorem 3.6 is more complex than in Theorem 3.2. However, the principal ideas are identical and we start again with sequences (Kn )n∈N , (Mn )n∈N tending to infinity and define for l = 1, 2



(l)

%n,m (s) = n max{t L

(l) (l)

(l)

in (s)+Vn,m (s)−1,n

−t



(l)

% n,m (s) = n t R

,t

(3−l) (3−l)

in

(3−l)

(s)+Vn,m (s)−1,n



(3−l) (3−l)

(3−l)

(s)+Vn,m (s)−1,n

in

}

,

(3−l) (3−l)

in

(3−l)

(s)+Vn,m (s),n

− min{t

 %n,m (s) = nκn,m (s)I (1) L (1)

(l) (l)

(l)

in (s)+Vn,m (s),n (1)

in (s)+Vn,m (s),n

 % n,m (s) = n(1 − κn,m (s))I (1) R (1)

∩I

,t



(3−l) (3−l)

in

(3−l)

(s)+Vn,m (s),n

(2) (2)

(2)

in (s)+Vn,m (s),n

(1)

in (s)+Vn,m (s),n

∩I

(4.7)

 ,

(2) (2)

} ,

(2)

in (s)+Vn,m (s),n

 ,

m = 1, . . . , Mn , where κn,m (s) denotes a U[0, 1]-distributed random variable and (1) (2) (Vn,m (s), Vn,m (s)) is distributed according to

   (l)  (1) (2) P (Vn,m (s), Vn,m (s)) = (k1 , k2 )S = I (1)

in (s)+k1 ,n

×



Kn  k1 ,k2 =−Kn

 (l) I (1)

in (s)+k1 ,n

∩I

(l) (2) in (s)+k2 ,n

∩I

−1  ,

(l) (2)

in (s)+k2 ,n

 

k1 , k2 ∈ {−Kn , . . . , Kn }. (4.8)

(1)

(2)

Again the random variables κn,m (s) and (Vn,m (s), Vn,m (s)) are defined on the ) and are apart from the requirement (4.8)  F, P extended probability space (Ω, independent of F and of each other for different m and/or different s. See Figure 4.2 for an illustration. (l) (l) Here, first two intervals I (1) , I (2) are chosen in the observation in (s)+k1 ,n

in (s)+k2 ,n

scheme of X (1) , X (2) with probability proportional to the intersection of these two intervals and then a point in this interval is chosen accordingly to the uniform random variable κn,m (s). Similarly as in the univariate setting stated in Remark 4.1 it holds L

(1)

(1)

(2)

(2)

%n,m (s), R % n,m (s), L%n,m (s), R % n,m (s), L%n,m (s), R % n,m (s)) Znbiv (Un,m (s)) = (L biv %n,m (s) =: Z

4.1 The Results

117

for F -conditionally independent random variables Un,m (s), m = 1, . . . , Mn , which are uniformly distributed on the interval K &n



k1 ,k2 =−Kn

I

= t

(1) (1)

in (s)+k1 ,n

∩I

(1)

(1),−

(1),+

(2)

∨t

(1)

in (s)−Kn −1,n



(2) in (s)+k2 ,n (2) (2)

in (s)−Kn −1,n

(2)

,t

(3)

(1) (1)

in (s)+Kn ,n

∧t



(2) (2)

in (s)+Kn ,n

.

(4)

Let Un,(i,j),m , Un,(i,j),m , Un,(i,j),m , Un,(i,j),m , Un,(i,j),m denote i.i.d. standard normal distributed random variables which are defined on the extended probability space and independent of all previously introduced random variables. Using these random variables and the random variables from (4.7) we proceed similarly as in the univariate setting and define by



% biv Φ T,n,m (f ) =

(1)

1{|Δ(1) X (1) |>β|I (1) | ,|Δ(2) X (2) |>β|I (2) | }

(2)

i,n

i,j:ti,n ∨tj,n ≤T



(1)



(2)

i,n

(1)

(1)

× ∂1 f (Δi,n X (1) , Δj,n X (2) ) σ ˜n (ti,n , −) (1)

$

(1)

+σ ˜n (ti,n , +)

$

+

(1)

j,n

$

j,n

(1),−

%n,m (τ n )U L i,j n,(i,j),m

(1),+

% n,m (τ n )U R i,j n,(i,j),m

(1)

(1)

(1)

(1)

(1)

(2)

%n,m (τ n ) + (˜ % n,m (τ n )U (˜ σn (ti,n , −))2 L σn (ti,n , +))2 R i,j i,j n,(i,j),m (1)



(2)

(2)

(2)

n + ∂2 f (Δi,n X (1) , Δj,n X (2) ) σ ˜n (tj,n , −)˜ ρn (τi,j , −) (2)

(2)

n +σ ˜n (tj,n , +)˜ ρn (τi,j , +)

$

(1),−

%n,m (τ n )U L i,j n,(i,j),m

(1),+

% n,m (τ n )U R i,j n,(i,j),m

Ln,m (s)

(1) Ln,m (s) = 0

$

 n,m (s) R

 (1) R n,m (s) = 0

X (1) (1)

() = t (1)

in (s)−2−1,n

(2)

(1)

∨ t (2)

in (s)−1−1,n

X (2)

+ κn,m (s)|I (1)

(2)

in (s)−2,n

∩ I (2)

in (s)−1,n

() s (2) Ln,m (s)

(1)

|I (1)

in (s)−2,n

(2)

∩ I (2)

in (s)−1,n

|



 (2) R n,m (s)

(1) (2) biv n,m (s) for Vn,m = −2, Vn,m (s) = −1. Figure 4.2: Realization of Z

|

118

4 Estimating Asymptotic Laws



(2)

(2)

n n %n,m (τi,j + (˜ σn (tj,n , −))2 (1 − (˜ ρn (τi,j , −))2 )L ) (2)

(2)

n n % n,m (τi,j + (˜ σn (tj,n , +))2 (1 − (˜ ρn (τi,j , +))2 )R )

$ +

(2)

(2)

(2)

(2)

(2)

1/2

(3)

Un,(i,j),m

(2)

(4)

%n,m (τ n ) + (˜ % n,m (τ n )U (˜ σn (tj,n , −))2 L σn (tj,n , +))2 R i,j i,j n,(i,j),m

 ,

m = 1, . . . , Mn , random variables whose F -conditional distribution approximates (1) (2) n the X -conditional distribution of Φbiv T (f ). Here, we set τi,j = ti,n ∧ tj,n to shorten notation and ρ˜n (s, −), ρ˜n (s, +) are consistent estimators for ρs− , ρs which are formally introduced in Chapter 6. Analogously to the considerations in the univariate situation we define by

%biv % % biv Q T,n (f, α) = Qα ({ΦT,n,m (f )|m = 1, . . . , Mn }) %biv Again Q T,n (f, α)

an estimator for the X -conditional α-quantile of Φbiv T (f ).

% biv corresponds to the α-quantile of the empirical distribution of the Φ T,n,m (f ), m = 1, . . . , Mn . %biv Then under the upcoming condition Q T,n (f, α) converges to the X -conditional biv α-quantile Qbiv T (f, α) of ΦT (f ) which is defined as the (under the following condition unique) X -measurable [−∞, ∞]-valued random variable fulfilling

   biv  Φbiv  P T (f ) ≤ QT (f, α) X (ω) = α,

ω ∈ Ω.

Condition 4.4. Suppose Condition 3.5 is fulfilled, the set (1)

(2)

(1)

(2)

{s ∈ [0, T ] : |∂1 f (ΔXs , ΔXs )| + |∂2 f (ΔXs , ΔXs )| > 0} (1) (2)

is almost surely not empty, the set {s ∈ [0, T ] : |σs σs | = 0} is almost surely T a Lebesgue null set and it holds 0 Γbiv (s, {0})ds = 0. Further, let the sequence (bn )n∈N fulfil bn = O(n−γ ) for some γ ∈ (0, 1) and suppose that (Kn )n∈N and (Mn )n∈N are sequences of natural numbers converging to infinity with Kn /n → 0. Additionally: (i) For any P ∈ N and xp ∈ R6 , sp ∈ (0, T ), p = 1, . . . , P , with sp = sp for p = p it holds P     biv   (Z biv (sp ) ≤ xp ) > ε → 0 P(Z%n,1 P P (sp ) ≤ xp , p = 1, . . . , P |S) − p=1

as n → ∞, for all ε > 0.

4.1 The Results

119

(ii) It holds (|πn |T )γ˜ +ε = oP (n−˜γ ) for any γ˜ > 0, ε > 0, and we assume n Gn 2,0 (T ) = OP (1), G0,2 (T ) = OP (1). (iii) The volatility process σ is itself an R2×2 -valued Itˆ o semimartingale, i.e. a process of similar form as (1.1). The roles of the specific assumptions in Condition 4.4 are identical to the roles of the corresponding assumptions in Condition 4.2. Part (ii) is an additional structural assumption which is needed in the proof of Theorem 4.5. Under Condition 4.4 we obtain the following result. Theorem 4.5. Suppose Condition 4.4 is fulfilled and f fulfils the assumptions (1,1) (2,2) made in Theorem 3.6 with p1 , p2 < 1. Then we have for α ∈ [0, 1]

    √n(V (f, πn )T − B ∗ (f )T ) ≤ Q  %biv lim P T,n (f, α) F = α

n→∞

(4.9)

for any F ∈ X with P(F ) > 0. Similarly as in the univariate case Theorem 4.5 allows to construct an asymptotic symmetric confidence interval with level α for B ∗ (f )T via



−1/2 % biv %biv V (f, πn )T − n−1/2 Q QT,n (g, (1 + α)/2) . T,n (g, (1 + α)/2), V (f, πn )T + n (1,1)

The assumption p1

p p

f(p,p ) (x, y) = x y p (p − 1)xp y p



−2

(2,2)

, p2

< 1 is not very restrictive. For example in the case 

we have ∂11 f(p,p ) (x, y) = p(p − 1)xp−2 y p , ∂22 f(p,p ) (x, y) = (1,1)

and p1

(2,2)

, p2

(l,l) pl

can be chosen to be zero because we need

(l,l)

(l,l)

p, p ≥ 2. Hence, usually ≤ p3−l which yields pl However, this assumption is needed in the proof.

(l,l)

< 1 due to pl

(l,l)

+p3−l = 1.

Example 4.6. The assumptions on the observation scheme made in Conditions 4.2 and 4.4 are fulfilled in the setting of equidistant and synchronous observation (1) (2) times ti,n = ti,n = i/n. In this specific situation we have seen in Example 3.3 that (l)

(l)

(δ− (s), δ+ (s)) is distributed like (κ(l) (s), 1 − κ(l) (s)) for κ(l) (s) ∼ U [0, 1]. Hence, it holds Γuniv,(l) (s, {0}) = 0 for any s ∈ [0, T ] and because of (l)

(l)

Znbiv (s) = (0, 0, 0, 0, nδn,− (s), nδn,+ (s)) as derived in Example 3.7 the same holds true for Γbiv (s, {0}). (l) (l) (l) (l) Further (δˆn,1,− (sp ), δˆn,1,+ (sp )) is distributed like (κn,m (sp ), 1 − κn,m (sp )) for (l) (l) (l) any sp ∈ [0, T ] because of |I | = n−1 for any i ∈ N. Hence (δˆ (sp ), δˆ (sp )) i,n

(l)

(l)

n,1,−

n,1,+

has the same distribution as (δ− (s), δ+ (s)). Then Condition 4.2(i) is fulfilled

120

4 Estimating Asymptotic Laws

(l) (l) (l) (l) because (δˆn,1,− (sp ), δˆn,1,+ (sp )) and (δˆn,1,− (s p ), δˆn,1,+ (s p )) are by construction (l)

(l)

independent for sp = s p as κn,m (sp ) and κn,m (s p ) are independent. These arguments immediately yield that Condition 4.4(i) is fulfilled as well because of (l) (l) Znbiv (s) = (0, 0, 0, 0, nδn,− (s), nδn,+ (s)) as above. Here we observe that in the case of synchronous observations the situation in the bivariate setting is not more complicated than in the univariate setting which intuitively seems reasonable.  Example 4.7. Conditions 4.2 and 4.4 are also fulfilled in the setting of Poisson sampling; compare Definition 5.1. The properties Γuniv,(l) (s, {0}) = 0 and Γbiv (s, {0}) = 0 for any s ∈ [0, T ] follow from the considerations in Remark 5.11 (l) (l) and because (δˆn,1,− (s), δˆn,1,+ (s)) and Znbiv (s) are almost surely different from

0 ∈ R2 respectively 0 ∈ R4 for any n ∈ N. That Conditions 4.2(i) and 4.4(i) are fulfilled in the Poisson setting follows from Lemma 5.12. Further part (ii) of Condition 4.4 is fulfilled by (5.2) and Corollary 5.6.  (l)

(l)

Remark 4.8. Instead of choosing realizations (nδn,− (s ), nδn,+ (s )) respectively Znbiv (s ) by picking s uniformly from the intervals



t t

(l) (l) in (s)−Kn −1,n

,t

(1) (1)

in (s)−Kn −1,n



(l) (l) in (s)+Kn ,n

∨t

,

(2) (2)

in (s)−Kn −1,n

,t

(1) (1)

in (s)+Kn ,n

∧t



(2) (2)

in (s)+Kn ,n

we might also set









(l) (l) (l) (l) κn,m (s)), nδn,+ (˜ κn,m (s)) , δˆn,m,− (s), δˆn,m,+ (s) = nδn,− (˜

biv %n,m Z (s) = Znbiv (˜ κn,m (s))

for i.i.d. random variables κ ˜ n,m (s), m = 1, . . . , Mn which are uniformly distributed on the interval [s − εn , s + εn ] for some sequence (εn )n∈N with εn → 0 and nεn → ∞ as n → ∞, compare the requirement on Kn in Condition 4.2. In this chapter, I chose to discuss the approach where s is picked from a fixed number of observation intervals because the resulting procedure is easier to implement for practical computations. 

4.2 The Proofs Conditions 4.2(i) and 4.4(i) yield that the empirical laws on the sets (l)

(l)

{(δˆn,m,− (s), δˆn,m,+ (s))|m = 1, . . . , Mn }, {Znbiv (s)|m = 1, . . . , Mn }

4.2 The Proofs

121 (l)

(l)

converge to the laws of (δ− (s), δ+ (s)) and Z biv (s). The following lemma will be the key tool for showing that this convergence already implies the convergence of the % univ,(l) (g)|m = 1, . . . , Mn } and {Φ % biv empirical laws on {Φ T,n,m (f )|m = 1, . . . , Mn } T,n,m univ,(l)

to the X -conditional laws of ΦT (g) and Φbiv T (f ). First, we prove a more general statement which then can also be used in Chapters 7–9 for proving similar results in the applications part.  P

Lemma 4.9. Suppose we have An,p −→ Ap for F -measurable random variables An,p ∈ Rd , X -measurable random variables Ap ∈ Rd , and let Sp ∈ [0, T ], p = 1, . . . , P , p ∈ N, be almost surely distinct X -measurable random variables. For  p %n,m n ∈ N, p = 1, . . . , P and s ∈ [0, T ] let Z (s), m = 1, . . . , Mn , be Rd -valued random variables which are independent of X , S-conditionally independent and whose S-conditional distributions are identical. Further let Z p (s), p = 1, . . . , P , be  Rd -valued random variables which are independent of X as well and suppose that  for any x = (x1 , . . . , xp ) ∈ Rd ×P and any ε > 0 it holds P     p   (Z p (sp ) ≤ xp ) > ε → 0 (4.10) P P(Z%n,1 P (sp ) ≤ xp , p = 1, . . . , P |S) − p=1

as n → ∞. Then it holds    Mn        1  ϕ((Ap , Z p (Sp ))p=1,...,P ) ≤ ΥX  > ε → 0 P P 1 − p   Mn  {ϕ((An,p ,Zn,m (Sp ))p=1,...,P )≤Υ} m=1

for any X -measurable random variable Υ, any ε > 0 and any continuous function  ϕ : R(d+d )×P → R such that the X -conditional distribution of the random variable ϕ((Ap , Z p (Sp ))p=1,...,P ) is almost surely continuous. Proof. First, note that (4.10) implies that   Mn  1   p     p P 1{Zn,m (sp )≤xp , p=1,...,P } − P(Z (sp ) ≤ xp , p = 1, . . . , P ) > ε (4.11)  Mn m=1 Mn         1  Z p (sp ) ≤ xp , p = 1, . . . , P S  > ε p P ≤P 1{Zn,m − n,1 (sp )≤xp , p=1,...,P } Mn m=1 2        p (sp ) ≤ xp , p = 1, . . . , P S − P  P  Z  Z p (sp ) ≤ xp , p = 1, . . . , P  > ε +P n,1 2

converges to zero as n → ∞ for any sp , xp , p = 1, . . . , P . In fact, Mn → ∞, the conditional Chebyshev inequality and dominated convergence ensure that the first p %n,m term vanishes asymptotically because the (Z (sp ))p=1,...,P are S-conditionally independent. The second term converges to zero by (4.10).

122

4 Estimating Asymptotic Laws

To shorten notation we set ζn =

Mn 1  1{ϕ((An,p ,Zn,m , p (Sp ))p=1,...,P )≤Υ} Mn m=1

    ϕ((Ap , Z p (Sp ))p=1,...,P ) ≤ ΥX . ζ=P The idea for the following steps is to approximate the function ϕ by piecewise constant functions, use (4.11) to prove the claim for those piecewise constant functions and to show that the convergence is preserved if we take limits. To formalize this approach let K > 0, set 

k (K, r) = {x ∈ Rd ×P |xi,p ∈ ((ki,p − 1)2−r K, ki,p 2−r K], i ≤ d , p ≤ P }, k = (k1 , . . . , kP ) ∈ Zd



×P

, r ∈ N, and define

ζn (K, r) =

Mn 1  1  Mn m=1 { k∈{−2r ,...,2r }d ×P



=

 k∈{−2r ,...,2r }d ×P

ζ(K, r) =



 k∈{−2r ,...,2r }d ×P

ϕ((Ap ,kp 2−r K)p≤P )1{(Z p

1{ϕ((Ap ,kp 2−r K)p≤P )≤Υ}

n,m (Sp ))p≤P )∈k (K,r)}

≤Υ}

Mn 1  1 p , Mn m=1 {(Zn,m (Sp ))p≤P )∈k (K,r)}

    (Z p (Sp ))p≤P ) ∈ k (K, r)X , 1{ϕ((Ap ,kp 2−r K)p≤P )≤Υ} P

where ϕ((Ap , kp 2−r K)p≤P ) equals ϕ((Ap , ·)p=1,...,P ) evaluated at the rightmost vertex of k (K, r). Using this notation it remains to show (|ζn − ζn (K, r)| > ε) = 0 lim lim sup lim sup P

K→∞

r→∞

(|ζn (K, r) − ζ(K, r)| > ε) = 0 lim P

n→∞

(|ζ(K, r) − ζ| > ε) = 0 lim lim sup P

K→∞

∀ε > 0,

(4.12)

∀K, ε > 0 ∀r ∈ N,

(4.13)

n→∞

r→∞

∀ε > 0.

(4.14)

Step 1. We start by showing (4.13). It holds (|ζn (K, r) − ζ(K, r)| > ε) P   Mn     1  P E 1 p ≤ Mn m=1 {(Zn,m (Sp ))p=1,...,P )∈k (K,r)} r r d ×P k∈{−2 ,...,2 }

        (Z p (Sp ))p=1,...,P ) ∈ k (K, r)X  > ε/(2r+1 + 1)d P X −P 

4.2 The Proofs

123

where each conditional probability vanishes almost surely as n → ∞ by (4.11) because the events p n,m (Sp ))p=1,...,P ) ∈ k (K, r)}, {(Z p (Sp ))p=1,...,P ) ∈ k (K, r)} {(Z

may be written as unions/differences of events of the form p n,m (Sp ))p=1,...,P ) ≤ vk,i (K, r)}, {Z p (Sp ))p=1,...,P ) ≤ vk,i (K, r)} {Z 

where vk,i (K, r), i = 1, . . . , 2d P , denote the vertices of the cuboid k (K, r). Note that conditioning on X here simply has the effect of fixing the Sp . (4.13) then follows by dominated convergence. Step 2. Next we show (4.12). It holds |ζn − ζn (K, r)| ≤

Mn 1  1 p d ×P ∨(A ) d×P } / / p p=1,...,P ∈[−K,K] Mn m=1 {(Zn,m (Sp ))p=1,...,P ∈[−K,K]   1 p + n,m {ϕ((An,p ,Z (Sp ))p=1,...,P )≤Υ}  k∈{−2r ,...,2r }d ×P

×

 − 1{ϕ((Ap ,kp 2−r K)p=1,...,P )≤Υ} 

Mn 1  1 p 1 d×P } . (4.15) Mn m=1 {(Zn,m (Sp ))p=1,...,P )∈k (K,r)} {(Ap )p=1,...,P ∈[−K,K]

The first term in (4.15) becomes arbitrarily small as first n → ∞ and then K → ∞, because for n → ∞ we obtain from (4.11) Mn 1  1 p d ×P } / Mn m=1 {(Zn,m (Sp ))p=1,...,P ∈[−K,K]

    P  −→ P (Z p (Sp ))p=1,...,P ∈ / [−K, K]d ×P X

as in Step 1, where the right-hand side vanishes as K → ∞. Denote the second term in (4.15) by ζn (K, r). Then it holds for δ > 0 ζn (K, r)

(4.16)

≤ 1{(An,p −Ap )p=1,...,P ≥δ} +

1 Mn

Mn  m=1

 p 1{(Zn,m (Sp ))p=1,...,P ∈[−K,K]d ×P }

p × 1{|ϕ((Ap ,Zn,m (Sp ))p=1,...,P )−Υ|≤ρ(K+δ,δ,2−r K)} 1{(Ap )p=1,...,P ∈[−K,K]d×P }

where ρ(K, a, b) is defined as sup

 (x,y),(x ,y  )∈[−K,K](d+d )×P :x−x  0 since An,p −→ Ap . Denoting the second summand in (4.16) by ζn

(K, r, δ) we further obtain ζn

(K, r, δ) ≤

Mn 1  Mn m=1

  k∈{−2r ,...,2r }d ×P

1{minx∈

k (K,r)

|ϕ((Ap ,xp )p=1,...,P )−Υ|≤ρ(K+δ,δ,2−r K)}

p × 1{(Zn,m (Sp ))p=1,...,P ∈k (K,r)} 1{(Ap )p=1,...,P ∈[−K,K]d×P }



≤

 k∈{−2r ,...,2r }d ×P

Mn  1   1 p Mn m=1 {(Zn,m (Sp ))p=1,...,P ∈k (K,r)}



+

 k∈{−2r ,...,2r }d ×P

    (Z p (Sp ))p=1,...,P ∈ k (K, r)X  −P

1{minx∈

k (K,r)

|ϕ((Ap ,xp )p=1,...,P )−Υ|≤ρ(K+δ,δ,2−r K)}

    (Z p (Sp ))p=1,...,P ∈ k (K, r)X 1{(A ) ×P d×P } (4.17) p p=1,...,P ∈[−K,K]

where the first sum vanishes for n → ∞ as shown in Step 1. Denote the second sum in (4.17) by ζn

(K, r, δ). Then we finally obtain     |ϕ((Ap , Z p (Sp ))p=1,...,P ) − Υ| ≤ 2ρ(K + δ, δ, 2−r K)X ζn

(K, r, δ) ≤ P which converges to zero because ϕ((Ap , Z p (Sp ))p=1,...,P ) possesses almost surely a continuous X -conditional distribution by assumption and because of lim lim sup ρ(K + δ, δ, 2−r K) = 0

δ→0

r→∞

for all K > 0 as ϕ is continuous. Hence altogether we have shown (|ζn (K, r)| > ε) lim lim sup lim sup lim sup P

K→∞

δ→0

r→∞

n→∞

for all ε > 0 which yields (4.12). Step 3. It holds  ζ(K, r) = P



  k∈{−2r ,...,2r }d ×P

   ϕ((Ap , kp 2−r K)p=1,...,P )1{(Z p (Sp ))p=1,...,P ∈k (K,r)} ≤ ΥX .

Hence 

((Z p (Sp ))p=1,...,P ) ∈ |ζ(K, r) − ζ| ≤ P / [−K, K]d ×P |X ) (|ϕ((Ap , Z p (Sp ))p=1,...,P ) − Υ| ≤ ρ˜(K, r, (Ap )p=1,...,P )|X ) +P

(4.18)

where ρ˜(K, r, (Ap )p=1,...P ) is defined as sup

 y,y  ∈[−K,K]d ×P :y−y  ≤2−r K

|ϕ((Ap , yp )p=1,...,P ) − ϕ((Ap , yp )p=1,...,P )|.

4.2 The Proofs

125

The first term on the right-hand side of (4.18) vanishes almost surely as K → ∞. Further it holds lim ρ˜(K, r, (Ap )p=1,...,P ) = 0 almost surely

r→∞



because y → ϕ((Ap , yp )p=1,...,P ) is uniformly continuous on [−K, K]d ×P for fixed ω. Using this result the second term in (4.18) vanishes almost surely as r → ∞ for any K > 0 because the X -conditional distribution of ϕ((Ap , Z p (Sp ))p=1,...,P ) is almost surely continuous by assumption. (4.14) then follows by dominated convergence. The following proposition shows that Condition 4.2 yields that the empirical % univ,(l) (g)|m = 1, . . . , Mn } converges to the X -conditional distribution on {Φ T,n,m univ,(l)

distribution of ΦT

(g).

Proposition 4.10. Suppose that Condition 4.2 is satisfied. Then it holds Mn         1  univ,(l) (g) ≤ ΥX  > ε → 0 (4.19) P 1 {Φ  univ,(l) (g)≤Υ} − P ΦT

Mn

T ,n,m

m=1

for any X -measurable random variable Υ and all ε > 0. (l)

Proof. Step 1. We denote by Sq,p , p ∈ N, an enumeration of the jump times (l)

(l)

of N (q) in [0, T ]. Here we choose the enumeration such that (Sq,p )p∈N is an increasing sequence which is possible because the process N (q) has finite jump activity. For using 4.9 we set



An,p = g (Δ

(l) (l)

(l)

in (Sq,p ),n

(l)

X (l) )1{|Δ(l)

(l)

(l) (l) in (Sq,p ),n

(l)

X (l) |>β|I



(l)

(l) | } (l) (l) in (Sq,p ),n

,σ ˜n (Sq,p , −), σ ˜n (Sq,p , +) ,



Ap = g (ΔX



(l) (l) Sq,p

), σ

(l) (l) Sq,p −

,σ

(l) (l) Sq,p



,

(l) (l) (l),− p %n,m Z (s) = δˆn,m,− (s), δˆn,m,+ (s), U (l)



n,in (s),m

(l)

(l)

(l)

(l)

Z p (s) = δ− (s), δ+ (s), U− (s), U+ (s)

,U





(l),+ (l)

n,in (s),m

,

and define ϕ via (l)

ϕ((Ap , Z p (Sq,p ))p=1,...,P ) =

P  p=1

g (ΔX

(l) (l)

Sq,p



) σ

(l) (l)

Sq,p −

(l)

(l)

(δ− (s))1/2 U− (s) + σ

(l) (l)

Sq,p

(l)

(l)



(δ+ (s))1/2 U+ (s) .

126

4 Estimating Asymptotic Laws P

By Condition 4.2(ii) An,p −→ Ap follows from Corollary 6.4 and (4.10) holds be(l),− (l),+ (l) cause of Condition 4.2(i) and because U (l) , U (l) respectively U− (s), n,in (s),m

n,in (s),m

(l) (l) (l) U+ (s) are standard normal distributed and independent of δˆn,m,− (s), δˆn,m,+ (s) (l)

(l)

respectively δ− (s), δ+ (s). Further the assumptions made in Condition 4.2 guar(l)

antee that the X -conditional distribution of ϕ((Ap , Z p (Sq,p ))p=1,...,P ) is almost surely continuous for q and P large enough and Lemma 4.9 then proves

 P

Mn     1     Y (P ) ≤ ΥX  > ε → 0  1{Y (P,n,m)≤Υ} − P

Mn

(4.20)

m=1

where Y (P, n, m) =

P 

g (Δ

p=1

(l) (l)

(l)

in (Sq,p ),n

X (l) )1{|Δ(l)

(l) (l) in (Sq,p ),n



X (l) |>β|I

(l) | } (l) (l) in (Sq,p ),n

(l) (l) (l) (l) (l),− × 1{S (l) ≤T } σ ˜n (Sq,p , −)(δˆn,m,− (Sq,p ))1/2 U (l)

(l)

n,in (Sq,p ),m

q,p

(l) (l) (l) (l) (l),+ +σ ˜n (Sq,p , +)(δˆn,m,+ (Sq,p ))1/2 U (l)



(l)

n,in (Sq,p ),m

Y (P ) =

P  p=1

g (ΔX



(l) (l) Sq,p

) σ

(l) (l)

(l)

Sq,p −

(l)

(l)

,

(l)

(δ− (Sq,p ))1/2 U− (Sq,p )

+σ

(l) (l) Sq,p

(l)

(l)

(l)

(l)



(δ+ (Sq,p ))1/2 U+ (Sq,p ) 1{S (l) ≤T } . q,p

Step 2. Next we prove lim lim sup

P →∞ n→∞

Mn   1   % univ,(l) (g) > ε = 0 P Y (P, n, m) − Φ T,n,m Mn

(4.21)

m=1

for all ε > 0. Denote by Ω(l) (P, r, q, n) the set on which there are at most P jumps of N (l) (q) in [0, T ], two different jumps of N (l) (q) are further apart than |πn |T (l) (l) and we have Δi,n (X (l) − N (l) (q)) ≤ 1/r for all i ∈ N with ti,n ≤ T . Obviously, lim lim sup lim sup P(Ω(l) (P, r, q, n)) = 1

q→∞ P →∞

n→∞

for any r > 0. On the set Ω(l) (P, r, q, n) we have

 

% univ,(l) |S 1Ω(l) (P,r,q,n) E |Y (P, n, m) − Φ T,n,m 

(l) ≤ E Kr (Δi,n (X (l) − N (l) (q)))2 1{|Δ(l) X (l) |>β|I (l) | } (l)

(l)

(l)

i:ti,n ≤T,p:Sq,p ∈Ii,n

i,n

i,n

4.2 The Proofs

127



(l) (l) (l) (l) (l),− × σ ˜n (ti,n , −)(δˆn,m,− (ti,n ))1/2 Un,i,m



≤

(l) E Kr (Δi,n (X (l) − N (l) (q)))2 1{|Δ(l) X (l) |>β|I (l) | }

(l)

i,n

i:ti,n ≤T

×

 

(l) (l) (l) (l) (l),+ +σ ˜n (ti,n , +)(δˆn,m,+ (ti,n ))1/2 Un,i,m S

1

bn

 (l)

(l)

(Δj,n X (l) )2

(l)

(l)

1/2

(l)

j =i:Ij,n ⊂(ti,n −bn ,ti,n +bn ]

(l)

i,n

(l)

 

(l)

× (δˆn,m,− (ti,n ) + δˆn,m,+ (ti,n ))1/2 S .

(4.22)

Using iterated expectations, Lemma 1.4 and inequality (2.77) we obtain  (l) E (Δi,n (X (l) − N (l) (q)))2 1{|Δ(l) X (l) |>β|I (l) | } i,n

i,n



1 × bn

(l)

(l)

(Δj,n X (l) )2

(l)

1/2   S

(l)

j=i:Ij,n ⊂(ti,n −bn ,ti,n +bn ]

 (l) ≤ E (Δi,n (X (l) − N (l) (q)))2 1{|Δ(l) X (l) |>β|I (l) | }  1 × bn

i,n

 (l)

i,n

1/2 (l) (Δj,n X (l) )2

(l)

(l)

j=i:Ij,n ⊂(ti,n −bn ,ti,n )

+

1 bn



(l)

(l)

(Δj,n X (l) )2

(l)

1/2   S

(l)

j=i:Ij,n ⊂(ti,n ,ti,n +bn ] (l)

≤ K(Kq |πn |T + (|πn |T )1/2− + eq )|Ii,n |. (l) (l) (l) (l) Hence as (δˆn,m,− (ti,n ) + δˆn,m,+ (ti,n ))1/2 is conditionally on S independent of all other random variables occuring in the conditional expectation we can bound (4.22) by

K(Kq |πn |T + (|πn |T )1/2− + eq ) ×



 

(l) (l) (l) (l) (l) |Ii,n |E (δˆn,m,− (ti,n ) + δˆn,m,+ (ti,n ))1/2 S

(l)

i:ti,n ≤T



≤ (Kq (|πn |T )1/2− + eq )

 

(l) (l) (l) (l) (l) |Ii,n |E (δˆn,m,− (ti,n ) + δˆn,m,+ (ti,n ))1/2 S

(l)

i:ti,n ≤T

Further by resorting the sum we obtain (Kq (|πn |T )1/2− + eq )

 (l)

i:ti,n ≤T

 

(l) (l) (l) (l) (l) |Ii,n |E (δˆn,m,− (ti,n ) + δˆn,m,+ (ti,n ))1/2 S

128

4 Estimating Asymptotic Laws



√ = (Kq (|πn |T )1/2− + eq ) n

(l)

|Ii,n |

(l) i:ti,n ≤T

Kn 

(l)

|Ii+k,n |3/2

k=−Kn

×



Kn 

(l)

|Ii+k ,n |

−1

k =−Kn



√ = (Kq (|πn |T )1/2− + eq ) n

(l)

× ≤ (Kq (|πn |T )



√

+ eq ) n

(l)

|Ii+k,n |

k=−Kn

i:ti,n ≤T

1/2−

Kn 

(l)

|Ii,n |3/2



Kn 

(l)

|Ii+k+k ,n |

−1

k =−Kn (l) |Ii,n |3/2

(l)

i:ti,n ≤T

×

0    (l) I

i+k,n

 

0 

 (l) I

i+k ,n

Kn −1   (l)  I +

i+k,n

k =−Kn

k=−Kn

= 2(Kq (|πn |T )

1/2−

Kn    (l)    I   −1 i+k ,n k =0

k=0

(l),n + eq )G3 (T ).

(4.23)

Hence (4.22) vanishes as first n → ∞ and then q → ∞ by Condition 3.1(i) and we have proven (4.21).  P

univ,(l)

Step 3. Using dominated convergence, we obtain Y (P ) −→ ΦT

(g) as

univ,(l) ΦT (g)

P → ∞. Also, as the X -conditional distribution of is continuous by Condition 4.2, for any choice of ε, η > 0 there exists δ > 0 such that

      univ,(l)      Φuniv,(l) (g) ± δ ≤ ΥX  > η < ε. P ΦT P (g) ≤ ΥX − P T Then it is easy to deduce that

  P  univ,(l)     Y (P ) ≤ ΥX −→  Φ P P (g) ≤ ΥX T holds for P → ∞. Step 4. For any ε > 0 we have Mn Mn  

 1    1  E 1{Y (P,n,m)≤Υ} − 1{Φ  univ,(l) (g)≤Υ}

Mn

 ≤E

m=1

Mn

1 

Mn

m=1

Mn

m=1

T ,n,m

1{|Y (P,n,m)−Φ  univ,(l) (g)|≥|Y (P,n,m)−Υ|} T ,n,m



(4.24)

4.2 The Proofs

 ≤E

129

Mn

1  

Mn

m=1

 1{|Y (P,n,m)−Φ  univ,(l) (g)|>ε} + 1{|Y (P,n,m)−Υ|≤ε} . T ,n,m

By (4.20) and dominated convergence we obtain

 E

Mn

1 

Mn

 (|Y (P ) − Υ| ≤ ε), 1{|Y (P,n,m)−Υ|≤ε} → P

(4.25)

m=1

where the right-hand side tends to zero as ε → 0 using dominated convergence again, because the X -conditional distribution of Y (P ) is continuous while Υ is X -measurable. By (4.21) we also have

 lim lim sup E

Mn

1 

P →∞ n→∞

Mn

m=1

 1{|Y (P,n,m)−Φ  univ,(l) (g)|>ε} = 0

(4.26)

T ,n,m

for all ε > 0. Thus, using (4.25) and (4.26), we obtain Mn        1 >ε =0 lim lim sup E 1{Y (P,n,m)≤Υ} − 1{Φ  univ,(l) (g)≤Υ} Mn P →∞ n→∞ T ,n,m m=1

(4.27) for all ε > 0. Step 5. Finally the claim (4.19) follows from (4.20), (4.24) and (4.27). Proof of Theorem 4.3. Step 1. We first prove (4.5). To this end note that we have for arbitrary ε > 0

 univ,(l)  univ,(l)  Q % P (g, α) > QT (g, α) + ε T,n  =P  ≤P

Mn  1 

Mn

m=1

Mn  1 

Mn

m=1

1{Φ  univ,(l) (g)>Quniv,(l) (g,α)+ε} > T ,n,m

T

Mn − (αMn  − 1)  Mn

1{Φ  univ,(l) (g)>Quniv,(l) (g,α)+ε} − Υ(α, ε) > (1 − α) − Υ(α, ε)



T ,n,m

 Φ with Υ(α, ε) = P T

univ,(l)



T

univ,(l)

(g) > QT

 

(g, α) + εX . Because the X -conditional

univ,(l)

distribution of ΦT (g) is continuous by Condition 4.2, we have Υ(α, ε) < 1 − α almost surely. Then it is easy to deduce

 univ,(l)  univ,(l)  Q % P (g, α) > QT (g, α) + ε → 0 T,n

130

4 Estimating Asymptotic Laws

using Proposition 4.10. Similarly we get

 univ,(l)  univ,(l)  Q % P (g, α) < QT (g, α) − ε → 0 T,n and combining these two convergences yields (4.5). Step 2. From Theorem 3.2 and (4.5) we obtain using Proposition B.7(i)



% Q T,n

univ,(l)

(g, α),

√

 L−s 

n(V (l) (g, πn )T − B (l) (g)T ) −→ QT

univ,(l)

univ,(l)

(g, α), ΦT



(g) .

From this property we obtain using the definition of X -stable convergence, compare (B.4), and (4.4)

 [ 1F 1 √ E { n(V (l) (g,π

n )T −B

(l) (g)

 univ,(l) (g,α)} T ≤QT ,n

]

 [1F 1 univ,(l) ] →E univ,(l) {Φ (g)≤Q (g,α)} T

 [1F P (Φ =E T

T

univ,(l)

univ,(l)

(g) ≤ QT

(g, α)|X )] = P(F )α

which is equivalent to (4.6). As for the proof of Theorem 4.3 to prove Theorem 4.5 we will first derive a proposition which will yield that Condition 4.4 implies the convergence of the % biv empirical distribution on the set {Φ T,n,m (f )|m = 1, . . . , Mn } to the X -conditional distribution of Φbiv T (f ). Proposition 4.11. Suppose that Condition 4.4 is satisfied. Then it holds Mn     1 P 1{Φ  biv

Mn

m=1

 



T ,n,m (f )≤Υ}



 Φbiv  >ε →0 −P T (f ) ≤ Υ X

(4.28)

for any X -measurable random variable Υ and all ε > 0. Proof. Denote by Sq,p , p ∈ N, an increasing sequence of stopping times which exhausts the common jump times of N (q) and define Y (P, n, m) = ×



P  p=1

1{|Δ(1)

(1) |>β|I (1) | ,|Δ(2) X (2) |>β|I (2) | } ip ,n jp ,n jp ,n

(1) (2) ∂1 f (Δip ,n X (1) , Δjp ,n X (2) ) (1)

$

(1)

+σ ˜n (tip ,n , +)

$

+

ip ,n X

(1)

(1)



(1) (1) σ ˜n (tip ,n , −)

$

1{Sq,p ≤T }

(1),− p ,jp ),m

%n,m (τpn )U L n,(i

(1),+ p ,jp ),m

% n,m (τpn )U R n,(i (1)

(1)

(1)

(1)



(2)

%n,m (τpn ) + (˜ % n,m (τpn )U (˜ σn (tip ,n , −))2 L σn (tip ,n , +))2 R n,(i

p ,jp ),m

4.2 The Proofs

131

(1)



(2)

(2)

(2)

+ ∂2 f (Δip ,n X (1) , Δjp ,n X (2) ) σ ˜n (tjp ,n , −)˜ ρn (τpn , −) (2)

(2)

+σ ˜n (tjp ,n , +)˜ ρn (τpn , +)



(2)

$

(1),− p ,jp ),m

%n,m (τpn )U L n,(i

(1),+ p ,jp ),m

% n,m (τpn )U R n,(i

(2)

%n,m (τpn ) + (˜ σn (tjp ,n , −))2 (1 − (˜ ρn (τpn , −))2 )L (2)

(2)

% n,m (τpn ) + (˜ σn (tjp ,n , +))2 (1 − (˜ ρn (τpn , +))2 )R

$

$

(2)

(2)

(2)

(2)

(2)

1/2

(3)

Un,(i

p ,jp ),m

(2)

(1)

(2)



(4)

%n,m (τpn ) + (˜ % n,m (τpn )U (˜ σn (tjp ,n , −))2 L σn (tjp ,n , +))2 R n,(i

+

(1)

p ,jp ),m

(2)

where we set ip = in (Sq,p ), jp = in (Sq,p ) and τpn = tip ,n ∧ tjp ,n . Further define P  

Y (P ) =

(1)

p=1 (1)

+ σSq,p

$

+

(2)



(1)

∂1 f (ΔXSq,p , ΔXSq,p ) σSq,p −



(1),+

(1)

(1)

(2)

(2)

+ σSq,p ρSq,p

+

$

(2)

(σSq,p − )2 L(1) (Sq,p ) + (σSq,p )2 R(1) (Sq,p )USq,p (1)

$

(1),−

L(Sq,p )USq,p

R(Sq,p )USq,p



(2)

+ ∂2 f (ΔXSq,p , ΔXSq,p ) σSq,p − ρSq,p −

+









(1),−

L(Sq,p )USq,p

(1),+

R(Sq,p )USq,p

(2)

(2)

(3)

(σSq,p − )2 (1 − (ρSq,p − )2 )L(Sq,p ) + (σSq,p )2 (1 − (ρSq,p )2 )R(Sq,p )USq,p (2)

(2)

(4)

(σSq,p − )2 L(2) (Sq,p ) + (σSq,p )2 R(2) (Sq,p )USq,p



1{Sq,p ≤T } .

p %n,m Step 1. By choosing An,p , Ap , Z (s), Z p (s) and the function ϕ appropriately, compare Step 1 in the proof of Proposition 4.10, we obtain the following convergence from Condition 4.4 using Lemma 4.9

 lim P

n→∞

Mn     1     Y (P ) ≤ ΥX  > ε = 0  1{Y (P,n,m)≤Υ} − P

Mn

(4.29)

m=1

for any P ∈ N. Step 2. Next we prove lim lim sup

P →∞ n→∞

Mn   1    % biv P Y (P, n, m) − Φ T,n,m (f ) > ε = 0. Mn

(4.30)

m=1

Denote by Ω(P, r, q, n) the set on which there are at most P common jumps of N (q) in [0, T ], two different jumps of N (q) are further apart than |πn |T and we

132

4 Estimating Asymptotic Laws (l)

(l)

have Δi,n (X (l) − N (l) (q)) ≤ 1/r, l = 1, 2, for all i ∈ N with ti,n ≤ T . Obviously, P(Ω(P, r, q, n)) → 1 for P, n, q → ∞ and any r > 0. On the set Ω(P, r, q, n) we have

 

 % biv E |Y (P, n, m) − Φ T,n,m (f )| F 1Ω(P,r,q,n)  ≤K 1{|Δ(1) X (1) |>β|I (1) | ,|Δ(2) X (2) |>β|I (2) | } (1)



(2)

(1)

i,n

(2)

i,j:ti,n ∨tj,n ≤T,p:Sq,p ∈Ii,n ∩Ij,n (1)

(2)

(1)

(1)

(2)

(2)

× |∂1 f (Δi,n X (1) , Δj,n X (2) )|˜ σn (i, n) + |∂2 f (Δi,n X (1) , Δj,n X (2) )|˜ σn (j, n)

$

$

i,n

j,n

j,n

(1)

(1)

(2)

(2)

%n,m + R % n,m + L%n,m + R % n,m )(τ n ) (L i,j

%n,m + R % n,m + L%n,m + R % n,m )(τ n ) (L i,j



× 1{I (1) ∩I (2) =∅} 1Ω(P,r,q,n) i,n

j,n



≤ Kr (1)

(2)

(1)

(2)

1{|Δ(1) X (1) |>β|I (1) | ,|Δ(2) X (2) |>β|I (2) | } i,n

i,j:ti,n ∨tj,n ≤T,p:Sq,p ∈Ii,n ∩Ij,n



(1,1)

(1)

× |Δi,n X (1) |1+p1

(1,1)

(2)

|Δj,n X (2) |p2

$

(2,2)

(1)

j,n

j,n

(1)

σ ˜n (i, n) (1)

(1)

(2)

(2)

%n,m + R % n,m + L%n,m + R % n,m )(τ n ) (L i,j

× + |Δi,n X (1) |p1

i,n

(2,2)

(2)

|Δj,n X (2) |1+p2

$

(2)

σ ˜n (j, n)

%n,m + R % n,m + L%n,m + R % n,m )(τ n ) (L i,j

×



× 1{I (1) ∩I (2) =∅} 1Ω(P,r,q,n) i,n

≤ Kr

j,n



(1) (2) i1 ,i2 :ti ,n ∨ti ,n ≤T 1 2



$

(l)

σ ˜n (il , n)

1{|Δ(1)

i1 ,n X

(1) |>β|I (1) | ,|Δ(2) X (2) |>β|I (2) | } i1 ,n i2 ,n i2 ,n

(l)

(l)

%n,m + R % n,m + L%n,m + R % n,m )(τ n ) (L i1 ,i2

l=1,2



(l,l)

(l)

× |Δil ,n (X − N (q))(l) |1+pl (l,l)

(l)

+ |Δil ,n X (l) |1+pl

(3−l)

(l,l)

|Δi3−l ,n X (3−l) |p3−l (l,l)

(3−l)



|Δi3−l ,n (X − N (q))(3−l) |p3−l 1{I (l)

(3−l) i1 ,n ∩Ii3−l ,n =∅}

(4.31)

where we applied the arguments following (3.55) to bound ∂l f (. . .), l = 1, 2, and (l) (l) used the shorthand notation σ ˜ (l) (i, n) = ((˜ σ (l) (ti,n , −))2 + ((˜ σ (l) (ti,n , +))2 )1/2 . To distinguish increments over the overlapping interval parts and the increments over (3−l,l) the parts which do not overlap we denote by Δ(j,i),n X the increment of X over (l)

(3−l)

(3−l\l)

(3−l)

(l)

the interval Ii,n ∩ Ij,n and by Δ(j,i),n X the increment of X over Ij,n \ Ii,n (which might be the sum of the increments over two separate intervals). Then we

4.2 The Proofs

133

obtain using Muirhead’s inequality as in (2.25), iterated expectations (note that (l) σ ˜ (l) (il , n) does not contain the increment over Iil ,n ), Lemma (1.4) and inequality (2.77) (l,l) ε (l,l)

(l) (1,2) (1,2) E σ ˜n (il , n)|Δ(i ,i ),n (X − N (q))(l) |1+pl |Δ(i ,i ),n X (3−l) |p3−l 1 2 1 2 ε   × 1{|Δ(l) X (l) |>β|I (l) | } S

il ,n



(l)

(1,2) (X 1 ,i2 ),n

≤E σ ˜n (il , n) |Δ(i

il ,n

(l,l)

− N (q))(l) |2 ε−2/(1+pl

)

1{|Δ(l)

il ,n X

(l,l)   (1,2) X (3−l) |2 ε2/p3−l S 1 ,i2 ),n

(l) |>β|I (l) | } il ,n

+ |Δ(i



(1)

(2)

(l,l)

(l)

≤ K (Kq |Ii1 ,n ∩ Ii2 ,n | + |Iil ,n |1/2− + eq )ε−2/(1+pl (1)

)

(l,l)

+ Kε2/p3−l



(2)

× |Ii1 ,n ∩ Ii2 ,n |. (l,l)

(l,l)

Note that p3−l > 0 because we assumed pl (l,l) 1+pl

(l)

(l,l)

(l,l)

< 1 and pl

+ p3−l = 1. Ana(l,l)

(3−l)

|Δi3−l ,n (X − N (q))(3−l) |p3−l can be logously the terms with |Δil ,n X (l) | bounded by a similar expression. Hence the S-conditional expectation of the last (1) (2) bound in (4.31) where we replace the increments over Ii1 ,n , Ii2 ,n with increments (1)

(2)

over Ii1 ,n ∩ Ii2 ,n is less or equal than 

Kr θ(n, q, ε)

(1)

(2)

|Ii,n ∩ Ij,n |

(1) (2) i,j:ti,n ∨tj,n ≤T



E[

$

(l)

(l)

%n,m + R % n,m + L%n,m + R % n,m )(τ n )|S] (L i,j

l=1,2

For some random variable θ(n, q, ε) with lim lim sup lim sup P(|θ(n, q, ε)| > δ) = 0,

ε→0 q→∞

n→∞

∀δ > 0.

(4.32)

Further we obtain using a similar index change as in (4.23)

 (1)

(1)

(2)

√

n

 (1)

E[

(l)

(l)

%n,m + R % n,m + L%n,m + R % n,m )(τ n )|S] (L i,j

(1)

Kn 

(2)

|Ii,n ∩ Ij,n |

(2)

× |Ii+k1 ,n ∩ Ij+k2 ,n | √ √ = OP ( n|πn |T ) + n



k1 ,k2 =−Kn

(2)

i,j:ti,n ∨tj,n ≤T (1)

$

l=1,2

i,j:ti,n ∨tj,n ≤T

=



(2)

|Ii,n ∩ Ij,n |



Kn 

(1) (2) i,j:ti,n ∨tj,n ≤T

(2)

(1)

(2)

|Ii+k ,n ∩ Ij+k ,n | 1

k1 ,k2 =−Kn



(1)

|Ii+k1 ,n |1/2 + |Ij+k2 ,n |1/2



(1)

−1

2

(2)



(1)

(2)

|Ii,n |1/2 + |Ij,n |1/2 |Ii,n ∩ Ij,n |



134

4 Estimating Asymptotic Laws Kn 

×

(1)

k1 ,k2 =−Kn

≤ OP (

√

(2)

|Ii+k1 ,n ∩ Ij+k2 ,n |



Kn 

(1)

|Ii+k

k1 ,k2 =−Kn

(2)

 1 +k1 ,n

∩ Ij+k

 2 +k2 ,n

n n|πn |T ) + 4(Gn 2,1 (T ) + G1,2 (T ))

|

−1

(4.33)

where we used Kn 

(1)

k1 ,k2 =−Kn

(2)

|Ii+k1 ,n ∩ Ij+k2 ,n |



Kn 

(1)

|Ii+k

k1 ,k2 =−Kn

(2)

 1 +k1 ,n

∩ Ij+k

 2 +k2 ,n

|

−1

≤4

which can be shown similarly as the corresponding estimate in (4.23). By Condition √ n 3.5 it holds OP ( n|πn |T ) + 4(Gn 2,1 (T ) + G1,2 (T )) = OP (1), compare (3.59). Hence the part of the last bound in (4.31) which stems from overlapping interval parts vanishes in the sense of (4.32) by Lemma 2.15. It remains to discuss the part of the last bound in (4.31) that originates from non-overlapping interval parts. To this end consider the notation (l)

Y(i,j),n = (l)

Y(i,j),n =

1 bn

1 bn



(l,3−l)

(Δ(k,j),n X (l) )2

1/2

(l) (l) (l) k =i:Ik,n ⊂[ti,n −bn ,ti,n +bn ]



(l\3−l)

(Δ(k,j),n , X (l) )2

,

1/2 ,

(l) (l) (l) k =i:Ik,n ⊂[ti,n −bn ,ti,n +bn ]

which allows to separate increments in σ ˜ (l) (i, n) over parts of intervals that do (3−l) overlap with Ij,n and parts of increments over intervals that do not overlap. From the Minkowski inequality we then obtain (l)

(l)

σ ˜ (l) (i, n) ≤ Y(i,j),n + Y(i,j),n . (l)

Let us first consider the last bound in (4.31) with σ ˜ (l) (i, n) replaced by Y(i,j),n . Using iterated expectations, the Cauchy-Schwarz inequality and Lemma 1.4 we obtain (l,l) (l,l)

(l) (3−l\l) (l) E |Δil ,n (X − N (q))(l) |1+pl |Δ(i3−l ,il ),n X (3−l) |p3−l Y(il ,i3−l ),n

(l\3−l) (X l ,i3−l ),n

+ |Δ(i

(l,l)

(l)

≤ K|Iil ,n |(1+pl

)/2

(l,l)

− N (q))(l) |1+pl (3−l)

(l,l)

(l,l)   (3−l,l) (l) X (3−l) |p3−l Y(i ,i ),n S ,i ),n 3−l l l 3−l

|Δ(i

(3−l)

|Ii3−l ,n |p3−l /2 |Ii3−l ,n |1/2 .

(4.34)

Then the last bound in (4.31) where we only consider increments over non(l) overlapping interval parts with σ ˜ (l) (i, n) replaced by Y(i,j),n is by

$

(l)

(l)

%n,m + R % n,m + L%n,m + R % n,m )(τ n ) ≤ (L i1 ,i2



n|πn |T ,

(4.35)

4.2 The Proofs

135

the estimate (4.34) and an anologous inequality for the term where the roles of X and X − N (q) are switched bounded by √ √  n  n|πn |T n|πn |T  n K G (T ) ≤ K G2,0 (T ) + Gn (l,l) 0,2 (T ) , (4.36) 1+pl ,p3−l (bn )1/2 (bn )1/2 l=1,2

compare (3.59). Note that bn = O(n−γ ), γ ∈ (0, 1), yields √ n|πn |T = O(n1/2+γ/2 |πn |T ). (bn )1/2 Hence the last bound in (4.36) vanishes as n → ∞ because by Condition 4.4(ii) we n have n1/2+γ/2 |πn |T = oP (1) due to γ < 1 and Gn 2,0 (T ) = OP (1), G0,2 (T ) = OP (1). Finally we consider the last bound in (4.31) where we only consider increments (l) over non-overlapping interval parts and with σ ˜ (l) (i, n) replaced by Y(i,j),n . The S-conditional expectation of that expression is using the estimate (4.35), iterated expectations and the arguments used in (2.82) bounded by





n|πn |T K



(1) (2) ∨ti ,n ≤T 1 ,n 2

i1 ,i2 :ti

(3−l)

× |Ii3−l ,n | ≤

√

p

(l)

|Ii1 ,n |

1+p

(l,l) (l,l) +(1/2−)(2−(1+p )) l l 2

l=1,2

(l,l) (l,l) +(1/2−)(2−(1+p )) 3−l l 2 (1,1)

n(|πn |T )1/2+(1/2−)((1−p1

(2,2)

)∧(1−p2

1{I (l)

(3−l) i1 ,n ∩Ii3−l ,n =∅}

))/2



Gn

l=1,2

≤

√

(1,1)

n(|πn |T )1/2+(1/2−)((1−p1

(2,2)

)∧(1−p2

))/2 

(l,l)

1+pl

,p3−l

(T )

n Gn 2,0 (T ) + G0,2 (T )



where the last bound vanishes as n → ∞ by Condition 4.4(ii) because of  < 1/2 (l,l) and pl < 1. Combining all above arguments we have shown that the S-conditional expectation of the last bound in (4.31) vanishes which yields (4.30) using Lemma 2.15. Step 3. As in Step 3 in the proof of Proposition 4.10 we deduce

  P  biv     Y (P ) ≤ ΥX −→  ΦT (f ) ≤ ΥX P P

(4.37)

for P → ∞. Step 4. Finally (4.28) follows from (4.29), (4.30) and (4.37); compare Steps 4 and 5 in the proof of Proposition 4.10. Proof of Theorem 4.5. The proof of (4.9) based on Proposition 4.11 is identical to the proof of (4.6) based on Proposition 4.10.

5 Observation Schemes In this chapter, we discuss examples of observation schemes that yield random irregular and asynchronous observations. We will investigate their asymptotics and check whether they fulfil the assumptions made in Chapters 2–4. I chose to collect these results in a separate chapter instead of including them in Chapters 2–4 firstly because it turns out that their proofs are rather technical and require specific arguments unrelated to the arguments in the previous chapters. Further the results in this chapter are of interest also on their own as they provide deeper insights into the nature of specific observation schemes. The observation schemes we are investigating in this chapter are of the following (l) form: Let t0,n = 0 and set (l)

(l)

(l)

ti,n = ti−1,n + r(l) (n)−1 Ei,n , i ≥ 1,

(5.1)

(l)

where the Ei,n , i, n ∈ N, are i.i.d. random variables with values in [0, ∞) and

r(l) : R → [0, ∞) are positive functions with r(l) (n) → ∞ as n → ∞ for l = 1, 2. Using this construction, the observation times of X (l) stem from a renewal process where the waiting times are i.i.d. and decrease with some rate r(l) (n), l = 1, 2. Observation schemes as in (5.1) will in the following be called renewal schemes. The most prominent example for a member of this class of observation schemes (l) is the case where the Ei,n are exponentially distributed and r(l) (n) = n. In this setting we have the nice property that the observation scheme is stationary in time. The specific rate r(l) (n) = n is chosen for convenience. As this observation scheme functions as our major example throughout this work we devote it its own name in the following definition. Most of the upcoming results will only be proven for this more specific observation scheme. (l)

Definition 5.1. We will call the observation scheme (πn )n∈N given by t0,n = 0 and (l)

(l)

ti,n = ti−1,n +

1 (l) E , i ≥ 1, n i,n

(l)

where the Ei,n , i, n ∈ N, are i.i.d Exp(λl )-distributed random variables, λl > 0, for l = 1, 2 Poisson sampling. 

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 O. Martin, High-Frequency Statistics with Asynchronous and Irregular Data, Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics, https://doi.org/10.1007/978-3-658-28418-3_5

138

5 Observation Schemes

This observation scheme has been discussed frequently in the high-frequency statistics literature; see e.g. [23] and [8]. In the Poisson sampling scheme we have the following result for the decay of the moments of |πn |T as n → 0. This result is taken from Lemma 8 of [23] where a proof can also be found. Lemma 5.2. Suppose (πn )n∈N is the Poisson sampling scheme. Then it holds

E[(|πn |T )q ] = o(n−γ ) for any 0 ≤ γ < q and q ≥ 1. By Jensen’s and Markov’s inequalities Lemma 5.2 immediately yields (|πn |T )q = oP (n−γ )

(5.2) P

for any 0 ≤ γ < q, q ≥ 0 and in particular it shows |πn |T −→ 0 in the case of Poisson sampling.

5.1 The Results The following lemma is a kind of law of large numbers for moments of the observation interval lengths in the setting where the observation times are generated by renewal processes. The main difference to the classical law of large numbers is that we do not work with a fixed number of random variables at each stage n. But instead, we sum up transformations of random variables as long as their sum does not exceed some threshold t. Lemma 5.3. Let the observation scheme be a renewal scheme as in (5.1) and (l) suppose we have E1,1 ∈ L2 (Ω). Then it holds r(l) (n)−1





(l)



P

g r(l) (n)|Ii,n | −→

(l)

i:ti,n ≤t

(l)

E[g(E1,1 )] (l)

E[E1,1 ]

t,

t ≥ 0,

(l)

for all functions g : [0, ∞) → R with g(E1,1 ) ∈ L1 (Ω). In the specific setting of Poisson sampling, the statement in Lemma 5.3 reads as stated in the following corollary. Corollary 5.4. Let (πn )n∈N be the Poisson sampling scheme introduced in Definition 5.1 and let E (l) be an Exp(λl )-distributed random variable, λl > 0. Then it holds n−1





(l)



P

g n|Ii,n | −→ E[g(E (l) )]λl t,

(l)

i:ti,n ≤t

for all functions g : [0, ∞) → R with g(E (l) ) ∈ L1 (Ω).

t ≥ 0,

5.1 The Results

139

In the case of Poisson sampling we also obtain a corresponding result in the bivariate setting where we consider sums which contain observation intervals from both processes X (l) , l = 1, 2. The following law of large numbers holds for functions f : [0, ∞)3 → R evaluated at the lenghts of the rescaled observation intervals (1)

(2)

(1)

(2)

(n|Ii,n |, n|Ij,n |, n|Ii,n ∩ Ij,n |) (1)

(2)

and included in the sum whenever Ii,n ∩ Ij,n =  ∅. However, unlike in Corollary 5.4, we can in general not find a simple closed form for the limit. Lemma 5.5. Let (πn )n∈N be the Poisson sampling scheme introduced in Definition 5.1 and let f : [0, ∞)3 → R be a function which fulfils

 (1)



(1)

(2)

(1)

(2)



f |Ii,1 |, |Ij,1 |, |Ii,1 ∩ Ij,1 | 1{I (1) ∩I (2) =∅} ∈ L2 (Ω). (5.3)

(2)

i,1

i,j:ti−1,1 ∧tj−1,1 ∈(0,1]

j,1

Then the expression 1 n

 (1)

(2)



(1)

(2)

(1)

(2)



f n|Ii,n |, n|Ij,n |, n|Ii,n ∩ Ij,n | 1{I (1) ∩I (2) =∅} i,n

i,j:ti,n ∨tj,n ≤t

j,n

converges for n → ∞ in probability to a deterministic function which is linear in t. By applying Lemma 5.5 for certain monomials we obtain the convergence of the (l),n n functions Gp , Gn p1 ,p2 , Hk,m,p introduced in Section 2.2 in the Poisson setting. (l),n

Corollary 5.6. In the Poisson setting we obtain that the functions Gp (t), n Gn p1 ,p2 (t), Hk,m,p (t) defined in (2.39) converge in probability to deterministic linear functions for any p, p1 , p2 ≥ 0 and 0 ≤ k, m with k + m ≤ p. Example 5.7. Although in general the limits in Lemma 5.5 and Corollary 5.6 cannot be easily computed, the limit has been computed for certain specific functions. Proposition 1 in [23] e.g. yields P

Gn 2,2 (t) −→ n (t) H0,0,4

P

 2

−→

λ1



+

2 t, λ2

2 t. λ1 + λ 2

Further, in the case where they cannot be explicitly computed the limits can always (l),n n naturally be estimated by the observed variables Gp (t), Gn p1 ,p2 (t), Hk,m,p (t).  As a further corollary of Lemma 5.5 we obtain the convergence of the functions (l),[k],n [k],n [k],n Gp , Gp1 ,p2 , Hι,m,p introduced in Section 2.4 in the Poisson setting.

140

5 Observation Schemes

Corollary 5.8. In the case of Poisson setting we obtain that the functions (l),[k],n [k],n [k],n Gp (t), Gp1 ,p2 (t), Hι,m,p (t) defined in (2.39) converge in probability to deterministic linear functions for any k ∈ N, p, p1 , p2 ≥ 0 and ι, m ≥ 0 with ι+m ≤ p. Remark 5.9. The above results could probably be generalized to hold for more observation schemes than just mere Poisson sampling. One possible extension would be to prove them in the case of inhomogeneous Poisson sampling where (l) the observation times ti,n , i ∈ N, follow Poisson point processes on [0, ∞) with

intensity measure nΛ(l) , l = 1, 2; compare Chapter 2 in [45] for a definition of Poisson point processes. If the intensity measure Λ(l) is sufficiently regular an approximation of Λ(l) by measures with piece-wise constant Lebesgue-densities might then be sufficient to carry the previously proven results in the situation of Poisson sampling over to the case of inhomogeneous Poisson sampling. To this end note that Poisson sampling, as introduced in Definition 5.1, corresponds to the situation described above with time-constant intensity measures nλl λλ, l = 1, 2, where λλ denotes the Lebesgue measure on [0, ∞). Further for a second posssible extension, note that the proof of Lemma (5.5) relies on the stationarity of the Poisson process only through the property that observation times within intervals [m, m + 1], [m , m + 1] are independent and identically distributed. If we instead use intervals of the form [mκn , (m + 1)κn ] with κn → ∞ as n → ∞ it would be sufficient that the renewal scheme is almost (l) (l) stationary in the long range in the sense that (ti,1 )i=1,...,K and (ti,1 )i=M,...,M +K become independent and that their distributions ”converge” in a certain sense as M → ∞ for any K ∈ N. Such a property should be fulfilled for ”most” renewal (l) schemes due to the i.i.d.-property of the waiting times Ei,n . However proving such a result would require a lot more arguments and is beyond the scope of this work.

 The above lemmata and corollaries show that the assumptions made in Chapter 2 on the observation scheme are fulfilled in the setting of Poisson sampling. The following result further shows that Conditions 3.1 and 3.5 which were required to obtain central limit theorems in Section 3.1 for the non-normalized functionals from Section 2.1 are fulfilled in the Poisson setting as well. The result is stated in a more general way to also cover the conditions we need to make in Chapters 7-9 to derive central limit theorems for certain selected applications. (1)

(1)



(2)

Set W t = (Wt , ρt Wt + 1 − ρ2t Wt )∗ as in (3.23) such that W Brownian motion driving the process X (l) for l = 1, 2. (1)

(2)

(l)

is the

(3)

Lemma 5.10. Let d ∈ N and Zn (s), Zn (s), Zn (s) be Rd -valued random variables, which can be written as  (l) Zn(l) (s) = f (l) n(s − t (l)

in (s)−1,n

(l)

), n(t (l)

in (s),n

√ (l) − s), ( nΔi(l) (s)+j,n W )j∈[k−1] , n

5.1 The Results √ (l) ( nΔ (l)

in (s)+j,k,n

141 W

(l)

1{I (3−l)

 (l) Zn(3) (s) = f (3) n(s − t (l)

) (l) ∩I (l) =∅} i∈{0,...,k−1},j∈Z (3−l) i (s)+i,k,n i (s)+j,k,n n

n

(l)

in (s)−1,n

), n(t (l)

in (s),n

(1)

− s)

(2)



l=1,2



, l = 1, 2,

,

(1)

(2)

n(s − t (1) ∨ t (2) ), n(t (1) ∧ t (2) − s), in (s)−1,n in (s)−1,n in (s),n in (s),n    (l) (1) (2) (n|I (l) |)i∈[k] l=1,2 , (n|I (1) ∩ I (2) |)i,j∈[k] in (s)+i,n

in (s)+i,n

in (s)+j,n

with [k] = {−k, . . . , k} for Borel-measurable functions f (1) , f (2) , f (3) and a fixed k ∈ N. Then in the situation of Poisson sampling introduced in Definition 5.1 the integral

 [0,T ]P1 +P2 +P3

×

P2 

(2)



g(x1 , . . . , xP1 , x 1 , . . . , x P2 , x

1 , . . . , x P3 ) E

P1 

(1)

(1)

hp (Zn (xp ))

p=1 (2)

hp (Zn (x p ))

p=1

P3 

(3)



(3)







hp (Zn (x

p )) dx1 . . . dxP1 dx1 . . . dxP2 dx1 . . . dxP3

p=1

(5.4) converges for n → ∞ to



[0,T ]P1 +P2 +P3

×

P1  



g(x1 , . . . , xP1 , x 1 , . . . , x P2 , x

1 , . . . , x P3 )

(1)

hp (y)Γ(1) (dy)

p=1

×

P3  

P2  

(2)

hp (y)Γ(2) (dy)

p=1 (3)



hp (y)Γ(3) (dy)dx1 . . . dxP1 dx 1 . . . dx P2 dx

1 . . . dxP3

(5.5)

p=1 (l)

for all bounded continuous functions g : RP1 +P2 +P3 → R, hp : Rd → R and all P1 , P 2 , P 3 ∈ N. Note that the measures Γ(1) , Γ(2) , Γ(3) implicitly defined in Lemma 5.10 are probability measures on Rd which do not depend on xp , x p , x

p like in the general case in Conditions 3.1(ii) and 3.5(ii). This property arises because the Poisson process has stationary increments. The key idea in the proof of Lemma 5.10 is to observe that due to the stationarity of the increments of the Poisson process and the Brownian motion W the law (l) of Zn (s) depends on s only through the fact that all observation intervals are bounded to the left because all observation times are greater or equal than zero. (l) (l) Further Zn (s) has the same distribution as Z1 (ns) and the effect on the law of

142

5 Observation Schemes

(l)

Z1 (ns) from the property that the observation intervals are bounded to the left (l)

becomes asymptotically negligible as n → ∞. Hence Z1 (ns) asymptotically has for ns → ∞ the same distribution as the corresponding random variable which (l)

is constructed based on observation times and Brownian motions W which are naturally extended on (−∞, 0]. Using this extended observation scheme and extended Brownian motions allows to characterize the limiting distribution. The following remark shows that in the case of Poisson sampling the limiting laws Γuniv (s, dy) and Γbiv (s, dy) have no atoms and also that Γuniv (s, dy) and Γbiv (s, dy) have bounded moments. This property yields that because Γuniv (s, dy) and Γbiv (s, dy) by Lemma 5.10 do not depend on s the laws described by Γuniv (s, dy) and Γbiv (s, dy) for s ∈ [0, T ] also have uniformly bounded first moments as required in Conditions 3.1(ii) and 3.5(ii). The computations in the following remark are also valid for the conditions which will be made in Chapters 7–9 where we will also need that the limiting laws do not admit atoms. Remark 5.11. In the proof of Lemma 5.10 we obtained L

(l)

Zn (s)1Ω(l) (s) = Z (l) 1Ω  (l) (s) n

n

(l)

(l)

 n (s))n∈N with for a random variable Z ∼ Γ and sequences (Ωn (s))n∈N , (Ω (l)  (l)  Ωn (s) ↑ Ω, Ω (s) ↑ Ω. Hence if the limit law would admit a part which is singular n (l) to the Lebesgue measure, this would also be true for the law of the Zn (s) for n (l) sufficiently large. In the two choices of the functions f for obtaining Conditions (l) 3.1(ii) and 3.5(ii) it can be shown that Zn (s) has no atom. Hence in Condition 3.1(ii) and 3.5(ii) we obtain Γuniv,(l) (x, {0}) = 0 and Γbiv (x, {0}) = 0. (l) and (Zn(l) )n∈N Further, by Theorem 6.7 from [9] there exist random variables Z



defined on a probability space (Ω , F , P ) with (l)

(l)

(l)

L

(l)

(l) , (Zn )n∈N ) = (Z (l) , (Zn )n∈N ) (Z n(l) → Z almost surely. An application of Fatou’s lemma then yields and Z (l) (l) (l) E[Z (l) p ] = E [Z(l) p ] ≤ lim inf E [Zn (s)p ] ≤ sup E [Zn (s)p ] = sup E[Zn (s)p ] n∈N

n∈N

n∈N

for p ≥ 0. For the cases corresponding to Conditions 3.1(ii) and 3.5(ii) it can be shown that the supremum is finite for all p ≥ 0 and hence Γuniv,(l) (x, dy) and Γbiv (x, dy) have finite moments for any x ∈ [0, T ]. This yields the requirement of uniformly bounded first and second moments in Condition 3.1(ii) and 3.5(ii).  Next, we will show that the assumptions made on the observation scheme in Conditions 4.2(i) and 4.4(i) are fulfilled in the case of Poisson sampling. We again derive a more general result which then later can also be used for the applications in Chapters 7–9.

5.2 The Proofs

143

Lemma 5.12. We consider the situation given in Lemma 5.10. Further define %(l) (s) = Zn(l) (Un(l) (s)) where we have one of the following three cases Z n,1 (l)

Un (s) ∼ t (l)

Un (s) ∼ t (l)

Un (s) ∼ t

(1) (1) in (s)−Kn −1,n

(2) (2)

in (s)−Kn −1,n (1) (1)

in (s)−Kn −1,n

,t ,t

(1) (1) in (s)+Kn ,n

(2) (2)

in (s)+Kn ,n

∨t

 

, ,

(2) (2)

(5.6)

in (s)−Kn −1,n

(5.7) ,t

(1) (1)

in (s)+Kn ,n

∧t

(2) (2)

in (s)+Kn ,n



(5.8)

for some sequence (Kn )n∈N of natural numbers with Kn → ∞ and Kn /n → 0. Let the distributions Γ(l) (dy), l = 1, 2, 3, be continuous and let Z (l) (s) ∼ Γ(l) (s, dy) be random variables independent of S. Then it holds P     (lp )   (Z (l) (sp ) ≤ xp ) > ε → 0 (5.9) P(Z%n,1 P P (sp ) ≤ xp , p = 1, . . . , P |S) − p=1

for any x ∈ Rd×P , P ∈ N, and sp ∈ (0, T ), lp ∈ {1, 2, 3}, p = 1, . . . , P , with sp = sp for p = p as n → ∞, for all ε > 0. In the proof of Lemma 5.12 we obtain the following nice property of Poisson processes: Let (Nt )t≥0 be a Poisson process and let Un ∼ U [0, cn ], n ∈ N, be uniformly distributed random variables on [0, cn ] with cn → ∞ as n → ∞ which are independent of the Poisson process (Nt )t≥0 . Then (Nt−Un − NUn )t≥Un has asymptotically the same law as (Nt )t≥0 and this also holds for the σ(Nt : t ≥ 0)conditional laws of (Nt−Un − NUn )t≥Un . Hence taking a fixed realization of a Poisson process and starting it at a uniformly distributed independent random time over an increasing time interval asymptotically yields a Poisson process again. This property is due to the stationarity of the Poisson process and the asymptotic independence of increments of the Poisson process over intervals which are far apart in time. Remark 5.13. The statement in Lemma 5.12 can also similarly be proven for (l) Un (s) ∼ U [s − εn , s + εn ] where (εn )n∈N denotes some deterministic sequence of positive real numbers with εn → 0 and nεn → ∞; compare Remark 4.8. 

5.2 The Proofs Proof of Lemma 5.3. This proof is based on the proof of Proposition 1 in [23]. Set (l) (l) (l) (l) m1 = E[E1,1 ], mg = E[g(E1,1 )] and (l)

λ(l) (n) = r(l) (n)T /m1 .

144

5 Observation Schemes

Further define (l)

NT (n) =

 i∈N

1{t(l) ≤T } . i,n

Step 1. We prove (l)

P(|NT (n) − λ(l) (n)| > r(l) (n)1/2+ε ) → 0

(5.10) (l)

as n → ∞ for ε ∈ (0, 1/2); compare Lemma 9 in [23]. Denote ν (l) = V ar(E1,1 ).

The case ν (l) = 0 is simple, so let us assume ν (l) > 0 in the following. It holds (l) NT (n) − λ(l) (n)

λ(l) (n)+r (l) (n)1/2+ε

>r

(l)

(n)

λ(l) (n)+r(l) (n)1/2+ε ⇔

i=1



1/2+ε



⇔

(l)

|Ii,n | < T

i=1 (l) (Ei,n

(l) − m1 )

(λ(l) (n) + r(l) (n)1/2+ε )ν (l)


r(l) (n)1/2+ε ) → 0.

(5.11)

Analogously it follows (l)

P(NT (n) − λ(l) (n) < −r(l) (n)1/2+ε ) → 0 which together with (5.11) yields (5.10). Step 2. By the weak law of large numbers it holds λ(l) (n)

r(l) (n)−1







(l) 

(l)

g r(l) (n)Ii,n  =

i=1

λ (n)   (l)  P λ(l) (n) 1 T (l) g Ei,n −→ (l) mg . r(l) (n) λ(l) (n) m1 i=1

Hence it remains to prove

r

(l)

(n)

−1

(l)

λ(l) (n)∨NT (n)



(l)

i=λ(l) (n)∧NT (n)+1





(l) 

P

g r(l) (n)Ii,n  −→ 0

(5.12)

5.2 The Proofs

145

as n → ∞. Because of (5.10) it suffices to prove the convergence (5.12) restricted to the set (l)

Ω(l) (n, ε) = {|NT (n) − λ(l) (n)| ≤ r(l) (n)1/2+ε }. On this set (5.12) is bounded by

2r

(l)

(n)

(1/2+ε)−1



1 2r(l) (n)1/2+ε

λ(l) (n)+α(l) (n)1/2+ε 

  (l)  g E  1



i,n

Ω(l) (n,ε)

i=λ(l) (n)−r (l) (n)1/2+ε +1

which converges in probability to zero for ε < 1/2 as n → ∞ because the term in (l) parantheses converges by the law of large numbers in probability to E[|g(E1,1 |]. Proof of Lemma 5.5. Due to the fact that π1 and nπn have the same law we obtain 1 n

 (1)



(1)

(2)

1 n

L

(1)

(2)



i,n

i,j:ti,n ∨tj,n ≤t

=

(2)

f n|Ii,n |, n|Ij,n |, n|Ii,n ∩ Ij,n | 1{I (1) ∩I (2) =∅}



(1)

(2)



(1)

(2)

(1)

(2)

j,n



f |Ii,1 |, |Ij,1 |, |Ii,1 ∩ Ij,1 | 1{I (1) ∩I (2) =∅} i,1

i,j:ti,n ∨tj,n ≤nt

j,1

nt nt 1  = Ym + OP (n−1 ) n nt m=1

L

where = denotes equality in law and



Ym = (1)



(1)

(2)

(1)

(2)



f |Ii,1 |, |Ij,1 |, |Ii,1 ∩ Ij,1 | 1{I (1) ∩I (2) =∅}

(2)

i,1

i,j:ti−1,1 ∧tj−1,1 ∈(m−1,m]

j,1

m ∈ N. Because of the stationarity of the Poisson process, the sequence Ym , m ∈ N, is a stationary and by (5.3) square integrable time series. Because further Ym1 , Ym2 become asymptotically independent as |m1 − m2 | → ∞ it is possible to conclude (Ym1 , Ym2 ) → 0 as |m1 − m2 | → ∞. We then obtain by Theorem 7.1.1 in [10] which is a law of large numbers for stationary processes 1 n

 (1)

(2)



(1)

(2)

(1)

(2)



P

f n|Ii,n |, n|Ij,n |, n|Ii,n ∩ Ij,n | 1{I (1) ∩I (2) =∅} −→ tE[Y1 ].

i,j:ti,n ∨tj,n ≤t

Hence the limit is linear in t with slope E[Y1 ].

i,n

j,n

146

5 Observation Schemes (l),n

Proof of Corollary 5.6. The convergence of Gp follows from Lemma 5.4 because all moments of an exponentially distributed random variable are finite. Lemma n 5.5 yields the convergences of Gn p1 ,p2 and Hk,n,p if we can verify (5.3) for the corresponding functions f . Because of n 0 ≤ Hk,n,p (t) ≤ Gn k,p−k (t),

t ≥ 0,

it then suffices to check (5.3) for the function f(p1 ,p2 ) (x, y, z) = xp1 y p2 , p1 , p2 ≥ 0. Therefore observe that we obtain using the Cauchy-Schwarz inequality



(E[( (1)

(1)

(2)

|Ii,1 |p1 |Ij,1 |p2 1{I (1) ∩I (2) =∅} )2 ])2

(2)

i,1

i,j:ti−1,1 ∧tj−1,1 ∈(0,1]



≤ (E[(|πn |T )p1 +p2 ( (1)

1{I (1) ∩I (2) =∅} )2 ])2

(2)

i,j:ti−1,1 ∧tj−1,1 ≤1

i,1



≤ E[(|πn |T )2(p1 +p2 ) ]E[(

j,1

(1) (2) i,j:ti−1,n ∧tj−1,n ≤1

j,1

1{I (1) ∩I (2) =∅} )4 ]. i,1

(5.13)

j,1

Here the first expectation in (5.13) is finite due to Lemma 5.2. Further

 (1) (2) i,j:ti−1,1 ∧tj−1,1 ≤1

1{I (1) ∩I (2) =∅} = i,1

j,1

∞  

1{t(l)

l=1,2 i=1

i−1,1 ≤1}

(l)

−1

(1)

(2)

because for i ≥ 1 each observation ti,n ∈ [0, 1] except the first one t1,n ∧ t1,n (1)

(2)

creates a new pair (i, j) with Ii,1 ∩ Ij,1 = ∅. Then the second expectation in ∞ (5.13) is finite in the Poisson setting because there the sum i=1 1{t(l) ≤1} − 1 is i−1,n

Poisson-distributed with parameter λl and all moments of the Poisson-distribution are finite. Proof of Corollary 5.8. Reproducing the proof of Lemma 5.4 we also obtain the convergence of 1 n

 (1)

(2)

i,j:ti,n ∧tj,n ≤t



(1)

(2)

(1)

(2)



f n|Ii,k,n |, n|Ij,k,n |, n|Ii,k,n ∩ Ij,k,n | 1{I (1)

(2) i,k,n ∩Ij,k,n =∅}

whenever the sum is square integrable for n = 1. As for Corollary 5.6 it then remains to verify this square integrability condition for appropriate functions f which can be done similarly as in the proof of Corollary 5.6. (l )

Proof of Lemma 5.10. First we show that any two random variables Zn 1 (x1 ) and (l ) Zn 2 (x2 ) with l1 , l2 ∈ {1, 2, 3}, x1 , x2 ∈ [0, T ], become asymptotically independent

5.2 The Proofs

147

for x1 = x2 which yields that the expectation in (5.4) factorizes in the limit. Let x1 < x2 and define Ω− n (x1 ) = Ω+ n (x2 ) =



max t l=1,2

(l)

≤

(l) (l ) )+k,n in (maxl =1,2 t (l ) in (x1 )+k

x1 + x2 2

≤ min t

x1 + x2

, 2



(l) (l)

l=1,2 in (minl =1,2 t

+ Ωn (x1 , x2 ) = Ω− n (x1 ) ∩ Ωn (x2 ).

(l ) )−k−1,n (l ) in (x2 )−k−1

,

Here, Ωn (x1 , x2 ) describes the subset of Ω on which the set of intervals used for (l ) the construction of Zn 1 (x1 ) and the set of intervals used for the construction of (l2 ) Zn (x2 ) are separated by (x1 + x2 )/2. Then there exist measurable functions g1 , g2 such that (l1 )

(l )

Zn 1 (x1 )1Ω− (x1 ) = g1 ((Nn (t))t∈[0,(x1 +x2 )/2] , (W t n

(l ) Zn 2 (x2 )1Ω+ (x2 ) n

)t∈[0,(x1 +x2 )/2] )1Ω− (x1 ) , n

= g2 ((Nn (t) − Nn ((x1 + x2 )/2))t∈[(x1 +x2 )/2,∞) , (l2 )

(W t (1)

(2)

(l2 )

− W (x1 +x2 )/2 )t∈[(x1 +x2 )/2,∞) )1Ω+ (x2 ) n

(l)

where Nn (t) = (Nn (t), Nn (t))∗ , Nn (t) =



i∈N 1{t(l) ≤t} ,

l = 1, 2, denotes the

i,n

Poisson processes which create the stopping times. These identities yield that the (l ) random variables Zn 1 (x1 )1Ω− (x1 ) , 1Ω− (x1 ) are independent from the random n

(l )

n

variables Zn 2 (x2 )1Ω+ (x2 ) , 1Ω+ (,x2 ) because the processes W and Nn (t), have n n independent increments. Hence we get (l )

(l )

(l )

(l )

E[hp11 (Zn 1 (x1 ))hp22 (Zn 2 (x2 ))1Ω− 1 + ] n (x1 ) Ωn (x2 ) (l )

(l )

(l )

(l )

= E[hp11 (Zn 1 (x1 ))1Ω− (x1 ) ]E[hp22 (Zn 2 (x2 ))1Ω+ (x2 ) ] n

=

=

n

(l ) (l ) E[hp11 (Zn 1 (x1 ))1Ω− 1 + ] n (x1 ) Ωn (x2 )

E [ 1Ω+ ] n (x2 )

(l ) (l ) E[hp22 (Zn 2 (x2 ))1Ω+ 1 − ] n (x2 ) Ωn (x1 )

E [ 1Ω− ] n (x1 )

(l ) (l ) (l ) (l ) E[hp11 (Zn 1 (x1 ))1Ωn (x1 ,x2 ) ]E[hp22 (Zn 2 (x2 ))1Ωn (x1 ,x2 ) ]

E[1Ωn (x1 ,x2 ) ]

which is equivalent to (l )

(l )

(l )

(l )

E[hp11 (Zn 1 (x1 ))hp22 (Zn 2 (x2 ))|Ωn (x1 , x2 )] (l )

(l )

(l )

(l )

= E[hp11 (Zn 1 (x1 ))|Ωn (x1 , x2 )]E[hp22 (Zn 2 (x2 ))|Ωn (x1 , x2 )].

(5.14)

148

5 Observation Schemes

Further we obtain using Markov’s inequality c + c P(Ωn (x1 , x2 )c ) ≤ P(Ω− n (x1 ) ) + P(Ωn (x2 ) )  (l) = P max t (l) (l ) l=1,2 in (maxl =1,2 t

+P

 x1 + x2

≤ 2Kλ1 ,λ2

2

(l ) in (x1 )+k

> min t

)+k,n

>

x1 + x2  2



(l) (l)

l=1,2 in (minl =1,2 t

(l ) )−k−1,n (l ) in (x2 )−k−1

1/n |x1 − x2 |

for a generic constant Kλ1 ,λ2 which yields

P(Ωn (x1 , x2 )) ≥ 1 −

K/n . |x1 − x2 |

(5.15)

Denote P = P1 + P2 + P3 , AT (ε) = {(x1 , . . . , xP ) ∈ [0, T ]P |∃i, j : |xi − xj | ≤ ε} and BT (σ) = {(x1 , . . . xP ) ∈ [0, T ]P : xσ(1) ≤ xσ(2) ≤ . . . ≤ xσ(P ) } where σ denotes a permutation of {1, . . . , P }. Using (5.14), (5.15), the inequality |E[X] − E[X|A]| ≤ 2K

1 − P(A) P(A)

which holds for any bounded random variable X ≤ K and any set A with P(A) > 0 (l) together with the boundedness of g, hp , l = 1, 2, 3, yields

  g(x1 , . . . , xP1 , x 1 , . . . , x P2 , x

1 , . . . , x

P3 ) (5.4) − [0,T ]P1 +P2 +P3

×

P2 P1     (1) (2)

E h(1) E h(2) p (Zn (xp )) p (Zn (xp )) p=1

×

p=1

  (3)







 E h(3) p (Zn (xp )) dx1 . . . dxP1 dx1 . . . dxP2 dx1 . . . dxP3 

P3 p=1

P⊗

≤ Kλλ  +

(AT (ε))

  (5.4) − 

σ∈SP

×E

BT (σ)\AT (ε)

P1 

(1) h(1) p (Zn (xp ))

P1 +P2

(2) h(2) p (Zn (xp ))

p=P1 +1

p=1

×

g(x1 , . . . , xP )

P p=P1 +P2 +1

(3) h(3) p (Zn (xp ))|

P  i=2

   Ωn (xσ(i−1) , xσ(i) ) dx1 . . . dxP 

5.2 The Proofs

+

 

 σ∈SP

BT (σ)\AT (ε)

3

×

149





m≤l

l=1 p=

 m 0. L (l) (l) For proving (5.16) observe that it holds Zn (x) = Z1 (nx). Since the Poisson processes generating the observation times in π1 and the Brownian motion (l)

(l)

W are independent stationary processes the law of Z1 (nx) depends on n only through the fact that all occurring intervals are bounded to the left by 0. Hence if Ω(n, m, x) denotes the set on which all intervals needed for the con(l) struction of Z1 (mx) are within [mx − nx, ∞), it follows that all members of the (l)

sequence (Z1 (mx)1Ω(n,m,x) ))m≥n have the same law. This yields because of P(Ω(n, m, x)) = P(Ω(n, n, x)), m ≥ n, and lim P(Ω(n, n, x)) = 1, x > 0,

n→∞ (l)

that the sequence (Z1 (nx))n∈N converges in law for x > 0. Hence the sequence (l)

(Zn (x))n∈N converges in law for x > 0 as well. By the stationarity of the processes the law of the limit, which we denote by Γ(l) , does not depend on x. Finally we obtain the uniform convergence in (5.16) because of P(Ω(n, n, x)) ≤ P(Ω(n, n, ε)) for x ≥ ε. (lp ) (lp ) %n,m %n,m Proof of Lemma 5.12. First, note that the variables Z (sp ) and Z (sp ) are (l )

p %n,m S-conditionally independent if we are on the set Ω(n, sp , sp ) on which Z (sp )

(l  )

p %n,m and Z (sp ) contain no common observation intervals. Without loss of generality let sp < sp . Using the Markov inequality and the notation (2.28) we get

 (m) (m ) P(Ω(n, sp , sp )c ) ≤ P max τn,+ ( max t (m )  +P



m =1,2 in

m=1,2

(sp )+Kn ,n

) ≥ sp + (sp − sp )/2

(m) (m ) min τ ( min t  ) ) m=1,2 n,− m =1,2 i(m (sp )−Kn −1,n n

≤ 2Kλ1 ,λ2



≤ sp + (sp − sp )/2



Kn /n (sp − sp )/2

for a generic constant Kλ1 ,λ2 . The latter tends to zero as n → ∞ because of (l )

(l  )

p p %n,m %n,m Kn /n → 0 by assumption. Hence, we may assume Z (sp ) and Z (sp ) to be S-conditionally independent, and it remains to prove (5.9) for P = 1. By assumption Z (l) (s) follows a continuous distribution. If we establish weak %(l) (s) to the (unconditional) convergence of the S-conditional distribution of Z n,1

one of Z (l) (s), then (5.9) follows from the Portmanteau theorem and dominated convergence. Hence it remains to prove (l)

(l)

L

S Zn (Un (s)) −→ Z (l) (s)

(5.17)

5.2 The Proofs

151

L

S where −→ denotes convergence of the S-conditional distributions. From here on we will give the proof only in the case (5.7). However the proofs in the cases (5.6) and (5.8) are up to some changes in notation identical. Note that by construction we have

(l)

(l)

(l)

L

(l)

(l)

(l)

Zn (Un (s)) = Zn (Un,Kn (ns)) = Z1 (U1,Kn (ns)), (l)

where Un,m (s) ∼ U [t valent to showing

(2) (2)

in (s)−m−1,n

(l)

,t

(2) (2)

in (s)+m,n

] and hence showing (5.17) is equi-

L

(1)

S Z1 (U1,Kn (ns)) −→ Z (l) (s)

(5.18)

because Z (l) (s) is independent of S. (l) As shown in the last paragraph of the proof of Lemma 5.10 the laws of Z1 (ns) (l) are equal to the law of Z (s) restricted to sets whose probability tends to 1 as (l) n → ∞. This property yields that as Z1 (ns) is a function of the observation (l)

intervals around time ns, and increments of the Brownian motions W over these intervals, the distribution of Z (l) (s) can be characterized as follows: Construct two (l) independent Poisson processes on (−∞, ∞) with intensities λl and jump times t˜i (l)

(l)

 )t∈(−∞,∞) with W  = 0. Here, a and two independent Brownian motions (W t 0 Poisson process (Nt )t∈R on (−∞, ∞) can be constructed from two independent (m) (1) Poisson processes (Nt )t≥0 , m = 1, 2, by setting Nt = Nt for t ≥ τ and (2)

Nτ −t = −Nt

where τ denotes the first jump time of N (1) . Then we use the (l)

(l)

function f from the defintion Z1 (ns) and apply it to the observation intervals (l) (l) over these intervals around time s. The random based on t˜i and increments W variable resulting from this construction has the same law as Z (l) (s). Based on these observations and the fact that the increments of the Brownian motions are merely independent standard normal random variables scaled by the interval length we conclude that for showing (5.18) it is sufficient to show that the S-conditional (l) (1) distribution of the intervals used in the construction of Z1 (U1,Kn (ns)) converges (l) to the intervals based on t˜i around time s. To formalize this idea we define for l = 1, 2 (l) (l) t˜0,n (s) = U1,Kn (ns), (l) (l) (l) (l) t˜k,n (s) = inf{ti,1 |ti,1 > t˜k−1,n (s)}, (l) t˜k,n (s)

=

(l) (l) sup{ti,1 |ti,1


2K + 1), this expression converges almost surely to 

 K−1  0 exp i j−k + vk+1 E j+k ) (i(v0 − v1 )E 0 )−1 E λ1 E (v−k E k=1



0 ) − exp(iv1 E 0 ) × exp(iv0 E   K−1

= E exp i ×

j−k + vk+1 E j+k ) (v−k E

k=1 ∞

 0

λ1 x

  K−1

= E exp i

 

exp(iv0 x) − exp(iv1 x) λ1 e−λ1 x dx i(v0 − v1 )x

j−k + vk+1 E j+k ) (v−k E

k=1



λ1 λ1 λ1 − iv0 λ1 − iv1

which is the characteristic function of a vector of 2K independent Exp(λ1 )distributed random variables. This yields (5.20). Analogously to (5.20) we obtain LS (2) (2) (2) (t˜k,n (s) − t˜k−1,n (s))k=−K+1,...,K −→ (Ek )k=−K+1,...,K ,

(5.22)

and finally (5.20) and (5.22) yield (5.19), because the Poisson process has stationary increments and because of the independence of the two processes we have that (1) (1) (2) (2) t˜k,n (s)−t˜k−1,n (s) and t˜k ,n (s)−t˜k −1,n (s) are asymptotically independent, because −1/2

dependency only occurs in the OP (Kn negligible.

)-term of (5.21) which is asymptotically

Part II

Applications

6 Estimating Spot Volatility (1)

(2)

Our goal in this chapter is to estimate the spot volatilities σs , σs at some specific time s ∈ [0, T ]. In addition to that we would like to estimate the spot correlation ρs between the two Gaussian processes C (1) and C (2) . If we allow σ to be discontinuous we are additionally interested in estimating the left limits (1) (2) σs− , σs− , ρs− . For finding suitable estimators note that we have



(l) (Δi,n C (l) )2

P



(1)

(l)

(σu )2 du,

l = 1, 2,

s

(l)

i:Ii,n ⊂(s,s+b]



s+b

−→

(1)

P

(2)

Δi,n C (1) Δj,n C (2) 1{I (1) ∩I (2) =∅} −→

(2)

i,n

i,j:Ii,n ∪Ij,n ⊂(s,s+b]

j,n



s+b

(1) (2)

ρu σu σu du s

for any b > 0 by Corollary 2.19 and Theorem 2.22 (compare also Example 2.23). Using the right-continuity of σ we further get 1 b 1 b

 

s+b

(l)

(l)

(σu )2 du → (σs )2 ,

l = 1, 2,

s s+b

(1) (2)

(1) (2)

ρu σu σu du → ρs σs σs s

as b → 0. Hence if we choose a deterministic sequence (bn )n∈N ⊂ (0, ∞) such that bn converges to zero sufficiently slowly we obtain 1 bn 1 bn



P

(l)

(l)

(Δi,n C (l) )2 −→ (σs )2 ,

l = 1, 2,

(l) i:Ii,n ⊂(s,s+bn ]



(1)

(2)

i,j:Ii,n ∪Ij,n ⊂(s,s+bn ]

(1)

P

(2)

(1) (2)

Δi,n C (1) Δj,n C (2) 1{I (1) ∩I (2) =∅} −→ ρs σs σs . i,n

j,n

In general we do not observe the Gaussian processes C (1) , C (2) but only the processes X (1) , X (2) . However, we will see that asymptotically this distinction here is not important because only jumps in the interval (s, s + bn ] enter the © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 O. Martin, High-Frequency Statistics with Asynchronous and Irregular Data, Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics, https://doi.org/10.1007/978-3-658-28418-3_6

158

6 Estimating Spot Volatility

estimation and “large” jumps vanish from the interval (s, s + bn ] as n → ∞ and bn → 0. Hence we can define consistent estimators via (l)

(˜ σn (s, +))2 = ρ˜∗n (s, +) =

1 bn



(l)

(Δi,n X (l) )2 ,

(l) i:Ii,n ⊂(s,s+bn ]



1

1

l = 1, 2,

(1) (2) σ ˜n (s, +)˜ σn (s, +) bn

(1)

(1)

(2)

Δi,n X (1) Δj,n X (2)

(2)

i,j:Ii,n ∪Ij,n ⊂(s,s+bn ]

× 1{I (1) ∩I (2) =∅} . i,n

j,n

In the small sample there still might be relatively large jumps of X (l) present (l) in the interval (s, s + bn ]. To avoid overestimation of σs we can use truncated increments to exclude unusually large increments which potentially include jumps from the estimation by using the following estimators (l)

(ˆ σn (s, +))2 = ρˆ∗n (s, +) =

1 bn

 (l) i:Ii,n ⊂(s,s+bn ]

(l)

(Δi,n X (l) )2 1{|Δ(l) X (l) |≤β|I (l) | } , i,n



1

1 (1) (2) b σ ˆ (s, +)ˆ σ (s, +) n n

n

(1)

l = 1, 2,

i,n

(1)

(2)

Δi,n X (1) Δj,n X (2)

(2)

i,j:Ii,n ∪Ij,n ⊂(s,s+bn ]

× 1{I (1) ∩I (2) =∅} 1{|Δ(1) X (1) |≤β|I (1) | } 1{|Δ(2) X (2) |≤β|I (2) | } i,n

j,n

i,n

i,n

j,n

j,n

for fixed constants β > 0 and  ∈ (0, 1/2); compare the discussion at the beginning of Section 2.3. (l) (l) Analogously we define σ ˜n (s, −), ρ˜∗n (s, −) and σ ˆn (s, −), ρˆ∗n (s, −) where we replace the interval (s, s + bn ] in the index of the sums with the interval (s − bn , s) such that we use increments over intervals to the left of s instead of intervals to the right of s for the estimation. For practical purposes one might require an estimator for ρ which lies in the interval [−1, 1]. Such an estimator can be easily obtained from the estimator introduced above by setting the estimator to 1 if it is larger than 1 and by setting it to −1 if it is less than −1. Therefore we define

















ρ˜n (s, +) = ρ˜∗n (s, +) ∧ 1 ∨ (−1),

ρ˜n (s, −) = ρ˜∗n (s, −) ∧ 1 ∨ (−1),

ρˆn (s, +) = ρˆ∗n (s, +) ∧ 1 ∨ (−1),

ρˆn (s, −) = ρˆ∗n (s, −) ∧ 1 ∨ (−1),

as proposed in Section 3.2 of [22]. Those estimators are also better than the unbounded estimators ρ˜∗n (s, +), ρ˜∗n (s, −), ρˆ∗n (s, +), ρˆ∗n (s, −) in the sense that they are always closer to the true values ρs , ρs− ∈ [−1, 1] than the unbounded estimators.

6.1 The Results

159

6.1 The Results The following two theorems show that the above constructions indeed yield consistent estimators and clarifies what it means for bn to converge to zero sufficiently fast. Theorem 6.1 states that if we work with arbitrary observation times that may (l) (l) also be endogenous the estimator σ ˜n (s, +) is consistent for σs , while Theorem 6.2 yields that if we restrict ourselves to exogenous observation times all above estimators are consistent. Theorem 6.1. Suppose Condition 1.3 holds and the sequence (bn )n∈N fulfils (l) (l) (l) (l) bn → 0, E[(δn (s))2 ]/b2n → 0 where δn (s) = sup{|Ii,n | : ti−1,n < s + bn }. Then it holds P

(l)

(l)

(˜ σn (s, +))2 −→ (σs )2 , l = 1, 2.

(6.1)

(l)

The reason why we need a bound on δn (s) here instead of a bound on |πn |T for some T > 0 will become clear in the proof of Theorem (6.1). (1) (2)

(1) (2)

Denote by Ω(s, −) = {σs− σs− = 0}, Ω(s, +) = {σs σs (1)

= 0} the sets on which

(2)

neither C nor C vanishes to the left respectively to the right of s. Only on these sets it is possible to estimate ρs respectively ρs− . Theorem 6.2. Suppose Condition 1.3 holds, the observation scheme is exogenous P

and we have bn → 0, |πn |T /bn −→ 0 for some deterministic T > s. Then it holds (l)

P

(l)

(˜ σn (s, −))2 −→ (σs− )2 ,

(l)

P

(l)

(ˆ σn (s, −))2 −→ (σs− )2 ,

(˜ σn (s, +))2 −→ (σs )2 , (ˆ σn (s, +))2 −→ (σs )2 ,

(l)

P

(l)

(l)

P

(l)

(6.2)

for l = 1, 2, P

ρ˜n (s, +)1Ω(s,+) −→ ρs 1Ω(s,+) ,

P

ρ˜n (s, −)1Ω(s,−) −→ ρs− 1Ω(s,−) ,

(6.3)

n and if we further assume (bn )−1 (Gn 1,1 (s + bn ) − G1,1 (s − bn )) = OP (1), see (2.39) for the definition of Gn , we also have 1,1 P

ρˆn (s, +)1Ω(s,+) −→ ρs 1Ω(s,+) , (l)

P

ρˆn (s, −)1Ω(s,−) −→ ρs− 1Ω(s,−) . P

(6.4)

(l)

Although the convergence (˜ σn (s, +))2 −→ (σs )2 has already been proven in Theorem 6.1 for the more general setting of possibly endogenous observation times P

we include it in Theorem 6.2 as the condition |πn |T /bn −→ 0 is slightly weaker than E[δ (l) (s)]/bn → 0, l = 1, 2.

160

6 Estimating Spot Volatility

It is not obvious whether the condition n (bn )−1 (Gn 1,1 (s + bn ) − G1,1 (s − bn )) = OP (1)

is really necessary to obtain the convergences of ρˆn (s, +) and ρˆn (s, −). However, by the method of proof used here, I didn’t manage to derive the needed estimates without it. Example 6.3. In the setting of Poisson sampling introduced in Definition 5.1 the assumptions made for Theorems 6.1 and 6.2 are fulfilled. We have bn → 0, P

(l)

E[δn (s)]/bn → 0 and |πn |T /bn −→ 0 by Lemma 5.2 for bn = n−α with α ∈ (0, 1).

n Further (bn )−1 (Gn 1,1 (s + bn ) − G1,1 (s − bn )) = OP (1) holds because n E[(bn )−1 (Gn 1,1 (s + bn ) − G1,1 (s − bn ))

= E[(nbn )−1 (G11,1 (n(s + bn )) − G11,1 (n(s − bn )))] is bounded by some constant; compare the proof of Corollary 5.6.



In some of the upcoming applications when testing for jumps we are not only interested in estimating the spot volatility and spot correlation at fixed deterministic (1) (2) times, but also at random times e.g. we would like to estimate σSp , σSp , ρSp where Sp = inf{s ≥ 0 : ΔXs  > 1} denotes the first time where X has a jump whose norm is larger than 1. This is also possible and the results are given in the following corollary. Corollary 6.4. Let τ be an (Ft )t≥0 -stopping time and suppose that we have (l)

E[(δn (T ))2 ]/b2n → 0, then it holds (l)

P

(l)

(˜ σn (τ, +))2 1{τ 0 almost surely which yields

P(τ ∈ {Spε |ε ∈ Q ∩ (0, ∞), p ∈ N}) = 1. Hence we also get P

(l)

(l)

σ ˜n (X, τ, −) −→ στ − ,

(l)

P

(1) (2)

κ ˜ n (X, τ, −) −→ ρτ − στ − στ −

from (6.34). Using these results we obtain (6.7) and (6.8) because (l)

(l)

P

(l)

(l)

P

σ ˜n (X ε , Spε , −) − σ ˆn (X ε , Spε , −) −→ 0, κ ˜ n (X ε , Spε , −) − κ ˆ n (X ε , Spε , −) −→ 0 follow as in the proof of Theorem 6.2.

7 Estimating Quadratic Covariation Historically the integrated volatility or realized volatility of the process X (l)



t 0

(l)

(σs )2 ds

and the integrated co-volatility of the processes X (1) and X (2)



t 0

(1) (2)

ρs σs σs ds

were among the first quantities that have been investigated in high-frequency statistics. They function as measures for how much a continuous process fluctuates or how much two continuous processes fluctuate “together“ and are therefore used to measure e.g. financial risk in terms of how volatile price processes are. The estimation of realized volatility dates back to [4] and [6]. While the spot volatilities and spot correlation investigated in Chapter 6 describe the size and dependence of instantaneous“ fluctuations, the integrated volatility and co-volatility are measures ” for the cummulative or average amount of fluctuations over a certain time interval. When working with stochastic processes that allow for jumps the quadratic variation process of the semimartingale X (l) which is defined by [X (l) , X (l) ]t =



t 0

(l)

(σs )2 ds +



(l)

(ΔXs )2

s≤t

and the covariation process of the processes X (1) and X (2) defined by [X (1) , X (2) ]t =



t 0

(1) (2)

ρs σs σs ds +



(1)

(2)

ΔXs ΔXs

s≤t

replace integrated volatility and co-volatility as measures for how much processes fluctuate (together). Hence it is of interest to find methods for estimating these quantities as well. The quadratic variation and covariation processes also play an important role in stochastic analysis. © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 O. Martin, High-Frequency Statistics with Asynchronous and Irregular Data, Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics, https://doi.org/10.1007/978-3-658-28418-3_7

176

7 Estimating Quadratic Covariation

Further in the setting of processes with jumps it is also of interest to quantify how large the contributions are that stem from the continuous parts respectively from the jump parts of X (1) and X (2) . Hence we are looking for estimators for the integrated (co-)volatility also if X is discontinuous and further we would like to find estimators for the jump parts



(l)

(ΔXs )2

and

s≤t



(1)

(2)

ΔXs ΔXs

s≤t

in the quadratic variation respectively the covariation process.

7.1 Consistency Results In the univariate and also in the bivariate but synchronous setting we have the following classical result; compare e.g. Theorem 23 in [41]. Theorem 7.1. Suppose Condition 1.3 is fulfilled. Then it holds



V (l) (g2 , πn )T =

P

(l)

(Δi,n X (l) )2 −→ [X (l) , X (l) ]T ,

l = 1, 2,

(l)

i:ti,n ≤T

where g2 (x) = x2 , x ∈ R. If the observation scheme is further synchronous it also holds V (f(1,1) , πn )T =



(1)

(2)

(1)

(2)

Δi,n X (1) Δj,n X (2) 1{I (1) ∩I (2) =∅} i,n

i,j:ti,n ∨tj,n ≤T

=



j,n

P

Δi,n X (1) Δi,n X (2) −→ [X (1) , X (2) ]T

i:ti,n ≤T

with f(1,1) (x, y) = xy, (x, y) ∈ R2 , where we used the notation from (2.1). When looking at the proofs of the convergences in Theorem 7.1 we observe that the continuous parts i.e. realized volatility and co-volatility of [X (l) , X (l) ]T and [X (1) , X (2) ]T originate from increments of the continuous part of X and the jump parts originate from increments of the jump part of X. The information on the continuous part is mostly captured in small increments of X while information on the jump part is contained in the comparably large increments. In Section 2.3 we introduced appropriate thresholds to separate comparably small respectively large increments and thereby separate the contributions of the continuous and jump parts in certain functionals. Using results from Section 2.3 we are able to obtain consistent estimators for the continuous and jump parts in the quadratic variation process. In the univariate setting Corollary 2.34 and part a) of Theorem 2.38 yield the following result.

7.1 Consistency Results

177

Theorem 7.2. Suppose Condition 1.3 is fulfilled and the observation scheme is exogenous. Then it holds (l)

P

(l)

P



V− (g2 , πn , (β, ))T −→

T 0



V+ (g2 , πn , (β, ))T −→

(l)

(σs )2 ds (l)

(ΔXs )2

s≤T

for β > 0 and  ∈ (0, 1/2). In the setting of asynchronous observations, a consistent estimator for the quadratic covariation process can be found from Theorem 3 in [8]. Further, we obtain similarly as in the univariate setting also in the bivariate setting consistent estimators for the continuous and jump parts in the quadratic covariation from Theorem 2.35 and part b) of Theorem 2.38. Theorem 7.3. Suppose Condition 1.3 is fulfilled and the observation scheme is exogenous. Then we have P

V (f(1,1) , πn )T −→ [X (1) , X (2) ]T .

(7.1)

If additionally it holds Gn 1,1 (T ) = OP (1), we also have P

V− (f(1,1) , πn , (β, ))T −→ P

V+ (f(1,1) , πn , (β, ))T −→



T 0



(1) (2)

ρs σs σs ds (1)

(2)

(7.2)

ΔXs ΔXs

s≤T

for β > 0 and  ∈ (0, 1/2). A result similar to (7.2) is stated in Corollary 3 of [34]. In that paper also a central limit theorem for an estimatorfor the integrated co-volatility is given  under the additional assumption that s≤T ΔXs β < ∞ holds almost surely for some β ∈ [0, 1) i.e. that the generalized Blumenthal-Getoor index is less than 1. Further there exists the following remarkable result which even holds for endogenous observation times which is due to [35] and based on results from [21]. Theorem 7.4. Suppose Condition 1.3 is fulfilled and the processes X (1) and X (2) are of finite jump activity on [0, T ] i.e. they almost surely have only finitely many jumps in the compact time interval [0, T ]. Then it holds P

V (f(1,1) , πn )T −→ [X (1) , X (2) ]T .

178

7 Estimating Quadratic Covariation

In the setting of finite jump activity processes further a consistent estimator for the realized the estimator for the covariation and for the co-volatility also lead to an estimator for



(1)

(2)

ΔXs ΔXs .

s≤T

Contrary to our approach they use truncated increments via a global threshold and not a local threshold depending on the length of the observation interval as in Section 2.3; compare the Notes at the end of Section 2.3.1.

7.2 Central Limit Theorems In this section, we construct central limit theorems accompanying the convergences in the previous section. We are going to restrict ourselves to finding central limit theorems in the setting of exogenous observation times. Central limit theorems have also been found for specific endogenous observation schemes, compare e.g. [18] and [47], but they are notoriously difficult to derive as in the case of endogenous observations the asymptotic variances also tend to depend on the covariance structure between the process and its components and the observation times. Further we are going to derive central limit theorems only for the estimators leading to the quadratic variation [X (l) , X (l) ]T respectively the covariation [X (1) , X (2) ]T at time T > 0. To be able to obtain feasibility of the central limit theorems in Section 7.3 we will develop stable central limit theorems; compare Chapter 3 and Appendix B. We will start by developing a central limit theorem for the estimator V (f(1,1) , πn )T in the univariate setting. To motivate the structure of the asymptotic variance we first consider the following simple toy example toy,(l)

Xt

(l)

= σ (l) Wt

+



ΔN (l) (q)s ,

t ≥ 0,

s≤t

for some q > 0. Here, the volatility σ (l) is assumed to be constant and it is assumed that X toy,(l) almost surely has only finitely many jumps in any compact time interval. Then on the set where any two jumps of X toy,(l) are further apart than |πn |T it holds V (l) (g2 , πn )T − [X toy,(l) , X toy,(l) ]T =



(l)

i:ti,n ≤T

(l)

(l)

(σ (l) Δi,n W (l) + Δi,n N (l) (q))2 − (σ (l) )2 T −

 s≤t

(ΔN (l) (q)s )2

7.2 Central Limit Theorems



= (σ (l) )2



179

(l)

(Δi,n W (l) )2 − |Ii,n |



(l)

i:ti,n ≤T



+ 2σ (l)

(l)

(l)

Δi,n W (l) Δi,n N (l) (q) + OP (|πn |T ).

(7.3)

(l) i:ti,n ≤T

The first term in this decomposition has mean zero and it holds

E

√



n(σ (l) )2



(l)

(Δi,n W (l) )2 − |Ii,n |

2   S

(l)

i:ti,n ≤T



= (σ (l) )4 n

(l),n

|Ii,n |2 E[(U 2 − 1)2 |S] = 2(σ (l) )4 G4

(T )

(l) i:ti,n ≤T

where U denotes a standard normal distributed random variable independent (l) of S. Further if we denote by in (s) the index of the interval characterized by (l) s ∈ I (l) , then for the second term in the decomposition (7.3) it holds in (s),n

√

n2σ (l)



(l)

L

(l)

Δi,n W (l) Δi,n N (l) (q) ≈ 2σ (l)

(l)

$

(l)

n|Ii,n (s)|Us ΔN (l) (q)s

s≤T

i:ti,n ≤T

where Us are i.i.d. standard normal random variables independent of F . Here we have asymptotic independence of the increments of the Brownian motion and the jumps which is intuitively due to the fact that increments over very small intervals asymptotically cannot have a huge influence on the global jump behaviour of the (l),n (l) process. If we assume that G4 (T ) converges in probability to some G4 (T ) and (l)

the (n|Ii,n (s)|)1/2 converge suitably in law to random variables η(s) it is possible to conclude  √  (l) n V (g2 , πn )T − [X toy,(l) , X toy,(l) ]T L−s

$

−→

(l)

2(σ (l) )4 G4 (T )U + 2σ (l)



η(s)Us ΔN (l) (q)s

s≤T

for i.i.d. standard normal random variables U, Us independent of F . The structure of the limit for general Itˆ o semimartingales is similar to the limit in the toy example. In particular the contribution of the processes B(q) and M (q) to the limit vanishes as q → ∞. However, the structure of the limit becomes more (l) complicated if we allow for time-varying volatility processes σs . We first recall the following notation (l)

(l)



(δn,− (s), δn,+ (s)) = s − t

(l) (l) in (s)−1,n

,t

(l) (l) in (s),n

−s



180

7 Estimating Quadratic Covariation (l)

(l)

δn,− (s)

δn,+ (s)

X (l) t

 (1) I (1)

(l) (l)

in (s)−1,n

in (s),n

 

t

(l) (l)

in (s),n

s (l)

(l)

Figure 7.1: Illustration of the terms δn,− (s) and δn,+ (s). introduced in Section 3.1, compare (3.6), to describe the distances of s to previous and upcoming observation times. For an illustration see Figure 7.1. The necessary assumptions required for obtaining a central limit theorem are collected in the following condition; compare Assumption 2 in [8]. Condition 7.5. Suppose Condition 1.3 is fulfilled, the observation scheme is exogenous and it further holds: (i) We have E[(|πn |T )q ] = o(n−α ) for any q ≥ α and 0 < α < q. (l),n

(ii) It holds G4

P

(l)

(t) −→ G4 (t), t ∈ [0, T ], for some increasing (possibly (l)

random) function G4 : [0, T ] → [0, ∞). (iii) The integral



T 0

g(x1 , . . . , xP )E

P



(l)

(l)



hp (nδn,− (s), nδn,+ (s)) dx1 . . . dxP

p=1

converges as n → ∞ to



T 0

g(x1 , . . . , xP )

P 

hp (y)ΓV AR,(l) (xp , dy)dx1 . . . dxP

p=1

for all bounded functions g : [0, T ]P → R, hp : [0, ∞)2 → R, p = 1, . . . , P , and any P ∈ N. Here ΓV AR,(l) (x, dy), x ∈ [0, T ], is a family of probability measures on [0, ∞)2 with uniformly bounded first moments. Part (iii) of Condition 7.5 is the specific assumption that is needed such that the (l) two components of n|I (l) | which correspond to the interval parts before and in (s),n

after s converge in the required sense. It is identical to part (ii) of Condition 3.1 and in particular it holds ΓV AR,(l) = Γuniv,(l) . Next we consider an extension ) on which we  F,  P of the probability space (Ω, F , P) to a probability space (Ω,

7.2 Central Limit Theorems

181

can define i.i.d. standard normal distributed random variables U , Us , s ∈ [0, T ], (l) (l) and random variables (δ− (s), δ+ (s)), s ∈ [0, T ], which are distributed according (l)

(l)

to ΓV AR,(l) (s, dy). Further, the random variables U , Us and (δ− (s), δ+ (s)), s ∈ [0, T ], are supposed to be independent of each other and of F . Using these newly introduced random variables we can state the central limit theorem for the estimator of the quadratic variation; compare Theorem 2 in [8]. Theorem 7.6. Assume that Condition 7.5 is fulfilled. Then we have the following X -stable convergence

 L−s √  √  (l) n V (g2 , πn )T − [X (l) , X (l) ]T −→ 2 '



+2

ΔXS (l)

(σ

p

(l) p:Sp ≤T

(l)

(l)

(l)

Sp −



T 0

(l)

(l)

(σs )4 dG4 (s)

(l)

(l)

1/2

(l)

U (l)

)2 δ− (Sp ) + (σSp )2 δ+ (Sp )US (l)

(7.4)

p

(l)

where (Sp )p∈N denotes an enumeration of the jump times of X (l) . To motivate the structure of the central limit theorem for the estimator of the covariation we again consider a toy example



Xttoy = σWt +

ΔN (q)s

s≤T

where the volatility matrix



σ (1) ρσ (2)

σ=



0 1 − ρ2 σ (2)



is constant and X has finite jump activity. Then on the set where any two jumps of X are further apart than 2|πn |T it holds V (f(1,1) , πn )T − [X toy,(1) , X toy,(2) ]T



=

(1)

(1)

(2)

i,j:ti,n ∨tj,n ≤T (2)

× (ρσ (2) Δj,n W (1) + − ρσ

(1)

(σ (1) Δi,n W (1) + Δi,n N (1) (q))

(1) (2)

σ

T−





(1) (2) i,j:ti,n ∨tj,n ≤T

(2)

(2)

1 − ρ2 σ (2) Δj,n W (2) + Δj,n N (2) (q))1{I (1) ∩I (2) =∅} i,n

ΔN

s≤T

= σ (1) σ (2)



(1)

(q)s ΔN

(1)

(2)

(q)s

(2)

Δi,n W (1) (ρΔj,n W (1) +



(2)

1 − ρ2 Δj,n W (2) )

j,n

182

7 Estimating Quadratic Covariation (1)

(2)



− ρ|Ii,n ∩ Ij,n | 1{I (1) ∩I (2) =∅} i,n



+ σ (2)

(1)

j,n

(1)

(2)

Δi,n N (1) (q)(ρΔj,n W (1) +



(2)

1 − ρ2 Δj,n W (2) )

(2)

i,j:ti,n ∨tj,n ≤T

× 1{I (1) ∩I (2) =∅} i,n j,n  (1) (1)

+ σ (1)

Δi,n W

(1)

(2)

Δj,n N (2) (q)1{I (1) ∩I (2) =∅} + OP (|πn |T ). i,n

(2)

i,j:ti,n ∨tj,n ≤T

j,n

(7.5) The first term in the above decomposition has mean zero and the S-conditional expectation of its square equals

E

√



n

(1)

(1)

(2)

Δi,n W (1) (ρΔj,n W (1) +



(2)

1 − ρ2 Δj,n W (2) )

(2)

i,j:ti,n ∨tj,n ≤T (1)

(2)



− ρ|Ii,n ∩ Ij,n | 1{I (1) ∩I (2) =∅} i,n

j,n

2    S

2 n 2 n Gn 2,2 (T ) + ρ H0,0,4 (T ) + ρ H (T ),

=

(7.6)

n compare (2.39) for the definition of the functions Gn 2,2 (t) and H0,0,4 (t), where

 n (t) = n H





l=1,2

(l) (3−l) (3−l) i,j,j  :ti,n ∨tj,n ∨tj  ,n ≤t

(l)

(3−l)

|Ii,n ∩ Ij,n

(l)

(3−l)

||Ii,n ∩ Ij  ,n |,

t ∈ [0, T ].

The calculations leading to (7.6) are presented in Section 7.4. Regarding the last two sums in (7.5) it holds similarly as in the univariate situation   √ (2)  (1) (2) (2) nσ Δi,n N (1) (ρΔj,n W (1) + 1 − ρ2 Δj,n W (2) )1{I (1) ∩I (2) =∅} (l)

i:ti,n ≤T

+

√

nσ (1)



(2)

Δj,n N (2) (q)

(2)

L

≈



i,n

(2)

j:tj,n ≤T

 (l)

j:tj,n ≤T

i:ti,n ≤T

(1)

Δi,n W (1) 1{I (1) ∩I (2) =∅} i,n

j,n

 ΔN (1) (q)s σ (2) ρ(ηn(1,2) (s))1/2 Us(1) + ρ(ηn(2\1) (s))1/2 Us(2)

s≤T



1 − ρ2 (ηn(1,2) (s) + ηn(2\1) (s))1/2 Us(3)   + ΔN (2) (q)s σ (1) (ηn(1,2) (s))1/2 Us(1) + (ηn(1\2) (s))1/2 Us(4) +



j,n

7.2 Central Limit Theorems

183 (1)

(2)

(3)

(4)

for i.i.d. standard normal random variables Us , Us , Us , Us , s ∈ [0, T ] and random variables (1,2)

ηn

(s) = n|I

(3−l\l)

ηn

(1) (1)

in (s),n

∪I

(2) (2)

in (s),n

|,



(s) = n

(3−l)

|Ij,n

(3−l) (3−l) (l) j =in (s):Ij,n ∪I (l)

=∅ in (s),n

\I

(l) (l)

in (s),n

|, l = 1, 2.

n n If we assume that Gn 2,2 (T ), H0,0,4 (T ), H (T ) converge in probability to some  G2,2 (T ), H0,0,4 (T ), H(T ) and that the random variables (1,2)

(ηn

(2\1)

(s), ηn

(1\2)

(s), ηn

(s))

converges suitably in law to random variables (η (1,2) (s), η (2\1) (s), η (1\2) (s)) it is possible to conclude

 √  n V (f(1,1) , πn )T − [X toy,(1) , X toy,(2) ]T L−s



 )] −→ (σ (1) σ (2) )2 [G2,2 (T ) + ρ2 H0,0,4 (T ) + ρ2 H(T +



ΔN (1) (q)s σ

s≤T

+



(2) 

(2)

(3)

(1)

(s))1/2 Us

U (2)

+ ρ(η (2\1) (s))1/2 Us (3) 

1 − ρ2 (η (1,2) (s) + η (2\1) (s))1/2 Us



(1,2)

+ ΔN (2) (q)s σ (1) (ηn (1)

(1,2)

ρ(ηn

1/2

(1)

(s))1/2 Us

(4) 

+ (η (1\2) (s))1/2 Us

(7.7)

(4)

where U, Us , Us , Us , Us , s ∈ [0, T ], denote i.i.d. standard normal random variables independent of F . To describe the limit for general Itˆ o semimartingales with potentially discontinuous volatility process σs we need to introduce some additional notation. Using the notation from (2.28) we define the random variables   (1) (2) (l) (1) (2) (s) = n min{τn,− (s), τn,− (s)} − τn,− (min{τn,− (s), τn,− (s)}) , l = 1, 2, LCOV,(l) n   (l) (1) (2) (1) (2) (s) = n τn,+ (max{τn,+ (s), τn,+ (s)}) − max{τn,+ (s), τn,+ (s)} , l = 1, 2, RCOV,(l) n   (1) (2) (s) = n s − min{τn,− (s), τn,− (s)} , LCOV n   (1) (2) (s) = n max{τn,+ (s), τn,+ (s)} − s , RCOV n ZnCOV (s) = (LCOV,(1) (s), RCOV,(1) (s), LCOV,(2) (s), RCOV,(2) (s), LCOV (s), RCOV (s)). n n n n n n (7.8)

184

7 Estimating Quadratic Covariation

For an illustration see Figure 7.2. Note that these random variables differ from those introduced in (3.14); compare also Figure 3.2. This is necessary because for the function f(1,1) idiosyncratic jumps contribute in the central limit theorem in (7.7) while in Section 3.1 we only looked at functions f where ∂1 (x, y) = ∂2 (x, y) = 0 for any (x, y) ∈ R2 with |xy| = 0 and hence only common jumps contributed in the central limit theorem 3.6. The necessary assumptions required to obtain a central limit theorem for the estimator of the quadratic covariation are collected in the following condition; compare Assumption 4 in [8]. Condition 7.7. Suppose Condition 1.3 and 7.5(i) are fulfilled, the observation scheme is exogenous and it further holds: n n (i) The functions Gn 2,2 (t), H0,0,4 (t), H (t) converge pointwise in t ∈ [0, T ] in  probability to increasing functions G2,2 (t), H0,0,4 (t), H(t).

(ii) The integral



T 0

g(x1 , . . . , xP )E

P





hp (ZnCOV (xp )) dx1 . . . dxP

p=1

converges as n → ∞ to

 0

COV,(1)

Ln

P 

T

g(x1 , . . . , xP )

(s)

hp (y)ΓCOV (xp , dy)dx1 . . . dxP

p=1

LCOV (s) n

RCOV (s) n

COV,(1)

Rn

(s) = 0

X (1)  (2) I

(2)

in (s),n

 (1) I

 

(1)

in (s),n

 

X (2) COV,(2)

Ln

(s) = 0

LCOV (s) n

RCOV (s) n

COV,(2)

Rn

(s)

s COV,(l)

Figure 7.2: Illustration of the terms Ln LCOV (s), RCOV (s). n n

COV,(l)

(s), Rn

(s), l = 1, 2, and

7.2 Central Limit Theorems

185

for all bounded functions g : [0, T ]P → R, hp : [0, ∞)6 → R, p = 1, . . . , P , and any P ∈ N. Here ΓCOV (x, dy), x ∈ [0, T ], is a family of probability measures on [0, ∞)6 with uniformly bounded first moments. Unlike in the univariate setting here part (ii) of Condition 7.7 is slightly different from the corresponding condition in Condition 3.5 because the variables ZnCOV and Znbiv differ. To describe the limit in the upcoming central limit theorem we denote  VtCOV =

t 0

(1) (2)

(σs σs )2 dG2,2 (s) +



t 0

(1) (2)

(ρs σs σs )2 dH0,0,4 (s) +



t 0

(1) (2)

 (ρs σs σs )2 dH(s),

t ∈ [0, T ]. Further let Z COV (s) = (LCOV,(1) (s), RCOV,(1) (s), LCOV,(2) (s), RCOV,(2) (s), LCOV (s), RCOV (s))

be distributed according to ΓCOV (s, dy) and defined on an extended probability ). U , Us(1),− , Us(1),+ , Us(2) , Us(3) , Us(4) , s ∈ [0, T ], are i.i.d. standard  F, P space (Ω, normal distributed random variables defined on the extended probability space as (1),− (1),+ (2) (3) (4) well. The random variables Z COV (s) and U , Us , Us , Us , Us , Us are all independent of each other and independent of F . Using these newly introduced random variables we define

 

ZtCOV =

p:Sp ≤t

$

(2)

+ σ S p ρSp

$

+

$

+

(1)

(1),−

LCOV (Sp )USp

(1),+

(2)

(2)

(3)

(σSp − )2 (1 − (ρSp − )2 )LCOV (Sp ) + (σSp )2 (1 − (ρSp )2 )RCOV (Sp )USp (2)

(2)

(4)

(σSp − )2 LCOV,(2) (Sp ) + (σSp )2 R(2) (Sp )USp

(2)

$

$

(2)

RCOV (Sp )USp



(1)

+ ΔXSp σSp − +



ΔXSp σSp − ρSp −

(1)

$

(1),−

LCOV (Sp )USp

(1)

+ σSp

(1)

$



(1),+

RCOV (Sp )USp (2)

(σSp − )2 LCOV,(1) (Sp ) + (σSp )2 RCOV,(1) (Sp )USp



where (Sp )p∈N denotes an enumeration of the jump times of X. Using the processes VtCOV and ZtCOV we are able to state the following central limit theorem; compare Theorem 3 in [8]. Theorem 7.8. If Condition 7.7 is fulfilled we have the following X -stable convergence

 L−s √  n V (f(1,1) , πn )T − [X (1) , X (2) ]T −→ ΦCOV := (VTCOV )1/2 U + ZTCOV . (7.9) T

186

7 Estimating Quadratic Covariation

The notation of the continuous term in the limit of Theorem 7.8 differs slightly from the notation of the corresponding term Theorem 3 of [8]. I chose a slightly different notation here to be closer to the notation used in Section 2.2. Example 7.9. Conditions 7.5 and 7.7 are fulfilled in the case of Poisson sampling introduced in Definition 5.1. Condition 7.5 holds by Lemma 5.2. Further Condition  n (t) 7.5(ii) and 7.7(i) follow from Corollary 5.6. Note that the convergence of H does not directly follow from Corollary 5.6, but can be proven similarly. That Conditions 7.5(iii) and 7.7(ii) are fulfilled in the Poisson setting follows from Lemma 5.10; compare Examples 3.4 and 3.8.  Notes. In this section I mainly presented results from [8]. However, I stated slightly modified versions of their theorems which are also valid in the setting of discontinuous volatility processes σ and thereby extend the applicability of their central limit theorems.

7.3 Constructing Asymptotic Confidence Intervals To turn the central limit theorems (7.4) and (7.9) into feasible central limit theorems which can be used in statistical applications we have to estimate the asymptotic variances. First of all we unfortunately find that the results from Section 2.2 and Section 2.3 do not yield consistent estimators for the terms V AR,(l)

VT and

VTCOV



T

=2 0

(l)

(l)

(σs )4 dG4 (s), l = 1, 2,

although the limits in the results there resemble these terms. In-

deed Corollary 2.19 and Theorem 2.38 yield that the quantities V (l) V − (4, g4 , πn , (β, ))T (l)

and processes X

(l)

(4, g4 , πn )T

V AR,(l) 3VT /2,

converge to but only for continuous . Further the limits of the two functionals V (4, f(2,2) , πn )T and

V − (4, f(2,2) , πn , (β, ))T have some resemblance to VTCOV , but they do not include

 and these functionals converge only for continuous the integral with respect to H X as well. (l) However, we are able to estimate the functions G4 (t), G2,2 (t), H0,0,4 (t) and (l),n

n  n H(t) pointwise in t ∈ [0, T ] by the sums G4 (t), Gn 2,2 (t), H0,0,4 (t) and H (t). This fact is trivial from the definition of these functions. Further we know from

7.3 Constructing Asymptotic Confidence Intervals

187 (l)

Chapter 6 consistent estimators for the spot volatilities σt ρt , t ∈ [0, T ]. Next we can approximate T /r V AR,(l)

VT

(r) =



V AR,(l) VT



(l)

and spot correlations

by Riemann-sums of the form

(l)

(l)

(σ(k−1)/r )4 G4 (k/r) − G4 ((k − 1)/r)



k=1 V AR,(l)

and it holds VT

V AR,(l)

(r) → VT

T /r V AR,(l)

VT

(r, n) =



pointwise in ω as r → ∞. Further



(l)

(l),n

(˜ σn ((k − 1)/r, +))4 G4

(l),n

(k/r) − G4

((k − 1)/r)



k=1 V AR,(l)

yields a consistent estimator for VT

(r). Note that instead of the estimators

(l) σ ˜n (s, +)

(l)

for the spot volatility we could also use the estimators σ ˆn (s, +). Based on the above considerations there has to exist some sequence of positive real numbers (rn )n∈N with rn → ∞ as n → ∞ and V AR,(l)

VT

P

V AR,(l)

(rn , n) −→ VT

.

Similarly we obtain a sequence of random variables VTCOV (rn , n) which consistently estimates VTCOV . To further investigate these estimators we have to specify the (l)

choice of rn . However, this choice depends on the convergence rates of σ ˜n (s, −), (l) σ ˜n (s, +), ρ˜n (s, −), ρ˜n (s, +) as well as on the convergence rates of the functions (l),n n n G4 (t),Gn 2,2 (t), H0,0,4 (t), H (t) which are even themselves difficult to determine. V AR,(l)

For this reason we will not further investigate the estimators VT VTCOV (rn , n).

(rn , n) and

To estimate the asymptotic distribution of V AR,(l)

ZT

=2

'



(σ

(l) p:Sp ≤T

(l)

(l)

(l)

Sp −

(l)

)2 δ− (Sp ) + (σ

(l) (l)

Sp

(l)

(l)

)2 δ+ (Sp )US (l) ΔXS (l) p

p

and of ZTCOV we can use the bootstrap method introduced in Chapter 4. First V AR,(l)

we discuss the estimation of the law of ZT

for which in a first step we have

(l) (l) (δ− (s), δ+ (s)).

to estimate the law of To this end let (Kn )n∈N and (Mn )n∈N denote increasing sequences of natural numbers and define (l)

(l)

(δˆn,m,− (s), δˆn,m,+ (s)), m = 1, . . . , Mn ,

188

7 Estimating Quadratic Covariation (l)

(l)

as in (4.1). Using these estimators for realizations of (δ− (s), δ+ (s)) we define for β > 0 and  ∈ (0, 1/2)

% Φ T,n,m

V AR,(l)

% = (V T,n

V AR,(l) 1/2

)



Un,m + 2

(l)

Δi,n X (l) 1{|Δ(l) X (l) |>β|I (l) | }

(l)

$ ×

i,n

i:ti,n ≤T (l)

(l)

(l)

(l)

(l)

(l)

i,n

(l)

(l)

(˜ σn (ti,n , −))2 δˆn,m− (ti,n ) + (˜ σn (ti,n , +))2 δˆn,m,+ (ti,n )Un,i,m

where Un,m , Un,i,m are i.i.d. standard normal distributed random variables inde-

% V AR,(l) is a suitable pendent of all previously introduced random variables and V T,n V AR,(l)

estimator for VT

% Q T,n

. Further we denote by

V AR,(l)

%α ({Φ % (α) = Q T,n,m |m = 1, . . . , Mn }) V AR,(l)

AR % VT,n,m % the αMn -largest element of the set {Φ |m = 1, . . . , Mn }. Then Q T,n converges under appropriate conditions to the X -conditional α-quantile of

V AR,(l)

V AR,(l) 1/2

(VT

)

(α)

V AR,(l)

U + ZT

which is the limit in Theorem 7.6. The X -conditional α-quantile QV AR,(l) (α) is defined as the (under certain conditions unique) X -measurable random variable fulfilling

    (V V AR,(l) )1/2 U + Z V AR,(l) ≤ QV AR,(l) (α)X (ω) = α, P T T

ω ∈ Ω.

To be able to consistently estimate an asymptotic confidence interval we need the following structural assumptions. Condition 7.10. Suppose Condition 7.5 is fulfilled, it holds (l)

T 0

(l)

|σs |ds > 0 almost

surely and t → G4 (t) is strictly increasing on [0, T ]. Further, let the sequence P

(bn )n∈N ⊂ (0, ∞) fulfil bn → 0, |πn |T /bn −→ 0 and suppose that (Kn )n∈N and (Mn )n∈N are sequences of natural numbers converging to infinity and Kn /n → 0. Additionally: (i) It holds

 (l) (l)   P P((δˆn,1,− (sp ), δˆn,1,+ (sp )) ≤ xp , p = 1, . . . , P |S) −

P 

  ((δ (l) (sp ), δ (l) (sp )) ≤ xp ) > ε → 0 P − +

p=1

as n → ∞, for all ε > 0 and any x = (x1 , . . . , xP ) ∈ R2×P , P ∈ N, and sp ∈ (0, T ), p = 1, . . . , P .

7.3 Constructing Asymptotic Confidence Intervals

189

(ii) The volatility process σ (l) is itself an Itˆ o semimartingale i.e. a process of the form (1.1). Part (i) of Condition 7.10 yields that the empirical distribution on the random variables (l)

(l)

(δˆn,m,− (sp ), δˆn,m,+ (sp )), (l)

m = 1, . . . , Mn ,

(l)

converges to the distribution of (δ− (sp ), δ+ (sp )), while part (ii) is needed for the (l)

(l)

convergence of the estimators σ ˜n (Sp , −). Under Condition 7.10 we obtain the following result.

% Theorem 7.11. Suppose Condition 7.10 is fulfilled and it holds V T,n

V AR,(l)

P

−→

COV %T,n VT for some sequence of F -measurable random variables V , n ∈ N. Then we have for α ∈ [0, 1] V AR,(l)

 



 [X (l) , X (l) ]T ∈ C V AR,(l) (α)F = α lim P T,n

(7.10)

n→∞

V AR,(l)

for any F ∈ X with P(F ) > 0 where CT,n % [V (l) (g2 , πn )T − n−1/2 Q T,n

V AR,(l)

(α) is defined as

% ((1 + α)/2), V (l) (g2 , πn )T + n−1/2 Q T,n

V AR,(l)

((1 + α)/2)].

We proceed similarly to estimate an asymptotic confidence interval for the covariation [X (1) , X (2) ]T . Let (Mn )n∈N be an increasing sequence of natural numbers and define COV %n,m Zn,m (s) = (L

COV,(l)

% n,m (s), R

COV,(l)

%n,m (s), L

COV,(2)

% n,m (s), R

COV,(2)

%COV % COV (s), L n,m (s), Rn,m (s)),

m = 1, . . . , Mn , via COV Zn,m (s) = ZnCOV (κn,m (s)),

κn,m (s) ∼ U [s − bn , s + bn ],

(7.11)

where κn,m (s) is apart from the above property independent of F and (bn )n∈N is a sequence of non-negative real numbers with bn → 0, nbn → ∞; compare Remark 4.8. Further we define by σ (l) (s, −) =

1 bn



(l)

(Δi,n X (l) )2

 12

,

(l)

i:Ii,n ⊂(s−bn ,s−|πn |T )

σ (l) (s, +) = σ ˜ (l) (s, +) =

1

bn

 (l)

i:Ii,n ⊂(s,s+bn )

(l)

(Δi,n X (l) )2

 12

,

190

7 Estimating Quadratic Covariation (l)

(l)

estimators for σs− , σs

where σ (l) (s, −) is a slightly modified version of the

(l)

˜ (s, −). The estimators σ ˜ (l) (s, −), σ ˜ (l) (s, +) have been introduced estimator σ and discussed in Chapter 6. Using these random variables we define  COV  COV 1/2 Un,m Φ T,n,m = (VT,n )

 (1) + 1{I (1) ∩I (2) =∅} Δi,n X (1) 1{|Δ(1) X (1) |>β|I (1) | } (1)

(2)

i,j:ti,n ∨tj,n ≤T

i,n

j,n

i,n

i,n

  (1) (1),− n n × σ (2) ρn (τi,j , −) LCOV n,m (τi,j )Un,(i,j),m n (ti,n , −)˜  (1) (1),+ n n  COV + σ (2) ρn (τi,j , +) R n,m (τi,j )Un,(i,j),m n (ti,n , +)˜  (1) 2 n n + (σ (2) ρn (τi,j , −))2 )LCOV n (ti,n , −)) (1 − (˜ n,m (τi,j )

1/2 (1) (3) 2 n n  COV ρn (τi,j , +))2 )R Un,(i,j),m + (σ (2) n (ti,n , +)) (1 − (˜ n,m (τi,j )   (2) (1) COV,(2) (2) (1) (4) n n  COV,(2) + (σ n (ti,n , −))2 Ln,m (τi,j ) + (σ n (ti,n , +))2 R (τi,j )Un,(i,j),m n,m   (2) (2) (1),− COV n + Δj,n X (2) 1{|Δ(2) X (2) |>β|I (2) | } σ (1) n (tj,n , −) Ln,m (τi,j )Un,(i,j),m j,n j,n  (2) (1),+  COV n + σ (1) n (tj,n , +) Rn,m (τi,j )Un,(i,j),m   (1) (2) COV,(1) (1) (2) (2) n n  COV,(1) + (σ n (tj,n , −))2 Ln,m (τi,j ) + (σ n (tj,n , +))2 R (τi,j )Un,(i,j),m n,m

× 1{|Δ(1) X (1) |=max i,n

k:I

× 1{|Δ(2) X (2) |=max j,n

(1)

(1) (2) ∩Ij,n =∅ k,n

k:I

(2) (1) ∩Ii,n =∅ k,n

(1)

|Δk,n X (1) |} (2)

|Δk,n X (2) |}

(2)

(1),−

(1),+

(2)

(3)

n where τi,j = ti,n ∧ tj,n and Un,m , Un,(i,j),m , Un,(i,j),m , Un,(i,j),m , Un,(i,j),m , (4)

Un,(i,j),m are i.i.d. standard normal distributed random variables independent COV %T,n of all previously introduced random variables and V is a suitable estimator

% COV for VTCOV . Here, the structure of the second part in Φ T,n,m is fundamentally % biv different from the structure of the estimator Φ T,n,m (f ) used in Chapter 4 because

% COV for Φ T,n,m idiosyncratic jumps also have to be estimated as they are contained in the variable ZTCOV which is part of the limit ΦCOV . Therefore summands T (1)

(1)

(2)

(2)

are included whenever |Δi,n X (1) | > β|Ii,n | or |Δj,n X (2) | > β|Ij,n | is fulfilled. (1)

(1)

The indicator over the set |Δi,n X (1) | = maxk:I (1) ∩I (2) =∅ |Δk,n X (1) | ensures that k,n

(2)

j,n

no increment Δj,n X (2) is included more than once in the sum and the indicator (2)

(2)

over the set |Δj,n X (2) | = maxk:I (2) ∩I (1) =∅ |Δk,n X (2) | ensures that no increment k,n

i,n

7.3 Constructing Asymptotic Confidence Intervals

191

(1)

Δi,n X (1) is included more than once. Note that the maxima are almost surely

unique if we assume that the volatility processes σ (1) , σ (2) do not vanish. Thereby we make sure that no jump (which is estimated by an increment exceeding the threshold) enters the sum multiple times. Further we use the estimator σ (l) (s, −) (3−l) instead of σ ˜ (l) (s, −) such that in the construction of σ (l) (ti,n , −) no increments (l)

(3−l)

over intervals Ij,n are used which overlap with Ii,n σ

(l)

(3−l) (ti,n , −)

whenever X

(l)

jumps in

(l) Ii,n , (l)

jump might also enter the estimation of σ

. Otherwise, as we evaluate

for co-jumps of X (l) and X (3−l) the (3−l)

and hence σ ˜ (l) (ti,n

(3−l) The estimator σ ˜ (ti,n , +) already has this desired % COV Based on the variables Φ T,n,m , m = 1, . . . , Mn , we (l)

, −) might diverge.

property. define via

%COV % % COV Q T,n (α) = Qα ({ΦT,n,m |m = 1, . . . , Mn }) an estimator for the X -conditional α-quantile QCOV (α) of (VTCOV )1/2 U + ZTCOV . To be able to consistently estimate an asymptotic confidence interval we need the following structural assumptions.

T

(1) (2)

Condition 7.12. Suppose Condition 7.5 is fulfilled, it holds 0 |σs σs |ds > 0 almost surely and the function t → G2,2 (t) is strictly increasing. Further, let the P

P

sequence (bn )n∈N ⊂ (0, ∞) fulfil bn → 0, |πn |T /bn −→ 0, n(|πn |T )2 /bn −→ 0 and suppose that (Mn )n∈N is a sequence of natural numbers converging to infinity. Additionally: (i) For any x ∈ R6×P , P ∈ N, and sp ∈ (0, T ), p = 1, . . . , P , it holds P     COV   (Z COV (sp ) ≤ xp ) > ε → 0 P P(Zn,1 P (sp ) ≤ xp , p = 1, . . . , P |S) − p=1

as n → ∞, for all ε > 0. (ii) The volatility process σ is itself an Itˆ o semimartingale, i.e. a process of the form (1.1). Under Condition 7.12 we are able to state the following result which yields an asymptotically valid confidence interval for [X (1) , X (2) ]T . COV P %T,n −→ VTCOV Theorem 7.13. Suppose Condition 7.12 is fulfilled and it holds V

COV %T,n for some sequence of F -measurable random variables V , n ∈ N. Then we have for α ∈ [0, 1]



 

COV  [X (1) , X (2) ]T ∈ CT,n lim P (α)F = α

n→∞

(7.12)

192

7 Estimating Quadratic Covariation

COV for any F ∈ X with P(F ) > 0 where CT,n (α) is defined as −1/2 % COV %COV [V (f(1,1) , πn )T − n−1/2 Q QT,n ((1 + α)/2)]. T,n ((1 + α)/2), V (f(1,1) , πn )T + n

Example 7.14. In the Poisson setting we have the convergences |πn |T /bn → 0 and n(|πn |T )2 /bn → 0 in probability whenever bn = O(n−α ) for some α ∈ (0, 1) by P

(l)

(5.2). Note also that |πn |T /bn −→ 0 implies nbn → ∞ because of |Ii,n | = OP (n−1 ) as n → ∞ and any i ∈ N, l = 1, 2. Further Conditions 7.10(i) and 7.12(i) are fulfilled by Lemma 5.12; compare also Remark 5.13 regarding the proof of Condition 7.12(i). 

7.4 The Proofs Proof of (7.6). We compute

√    (1) (1) (2) (2) E n Δi,n W (ρΔj,n W (1) + 1 − ρ2 Δj,n W (2) ) (1)

(2)

i,j:ti,n ∨tj,n ≤T

2    (1) (2)  − ρ|Ii,n ∩ Ij,n | 1{I (1) ∩I (2) =∅} S i,n j,n

   (1) (1) (2) (2) (1) (2)  = nE Δi,n W (ρΔj,n W (1) + 1 − ρ2 Δj,n W (2) ) − ρ|Ii,n ∩ Ij,n | (1)

(2)

i,j:ti,n ∨tj,n ≤T



×



(1)

(1) (2) i ,j  :t  ∨t  ≤T i ,n j ,n

(1)

(2)

Δi ,n W (1) (ρΔj  ,n W (1) +

(2)

− ρ|Ii ,n ∩ Ij  ,n |



(2)

1 − ρ2 Δj  ,n W (2) )



   × 1{I (1) ∩I (2) =∅} 1{I (1) ∩I (2) =∅} S i,n j,n i ,n j  ,n

    (1) (1) (2) (2) = nE Δi,n W (ρΔj,n W (1) + 1 − ρ2 Δj,n W (2) ) (1)

(2)

(1)

(2)

i,j:ti,n ∨tj,n ≤T i ,j  :t  ∨t  ≤T i ,n j ,n (1)

(2)

× Δi ,n W (1) (ρΔj  ,n W (1) + × 1{I (1) ∩I (2) =∅} 1{I (1) i,n

= (1 − ρ

2

+ nρ2 E

i ,n

j,n

)Gn 2,2 (T )



(2)

∩I  j ,n

(2)

(1)

(2)

(1)

(2)

1 − ρ2 Δj  ,n W (2) ) − ρ2 |Ii,n ∩ Ij,n ||Ii ,n ∩ Ij  ,n |    S =∅}





(1) (2) i,j:ti,n ∨tj,n ≤T

(1) (2) i ,j  :t  ∨t  ≤T i ,n j ,n



(1)

(2)

(1)



(2)

Δi,n W (1) Δj,n W (1) Δi ,n W (1) Δj  ,n W (1)

(1) (2) (1) (2)  − |Ii,n ∩ Ij,n ||Ii ,n ∩ Ij  ,n | 1{I (1) ∩I (2) =∅} 1{I (1) i,n

j,n

(2) ∩I  =∅} i ,n j ,n

   S .

(7.13)

7.4 The Proofs

193

The last equality holds because for terms including increments of W 2 only expect(2) (2) ations of those terms including Δj,n W (2) Δj  ,n W (2) with j = j and consequently (1)

(1)

also containing Δi,n W (1) Δi ,n W (1) for i = i do not vanish. Next, we discuss the second sum in (7.13). For the terms with i = i and j = j it holds

(1) (2) (1) (2) (1) (2) (1) (2)   E Δi,n W (1) Δj,n W (1) Δi ,n W (1) Δj  ,n W (1) − |Ii,n ∩ Ij,n ||Ii ,n ∩ Ij  ,n |S

(1) (2) (1) (2)   = E (Δi,n W (1) )2 (Δj,n W (1) )2 − |Ii,n ∩ Ij,n |2 S (1)

(2)

(1)

(2)

(1)

(2)

= 3|Ii,n ∩ Ij,n |2 + |Ii,n \ Ij,n ||Ii,n ∩ Ij,n | (1)

(2)

(2)

(1)

(1)

(2)

(2)

(1)

(1)

(2)

+ |Ii,n ∩ Ij,n ||Ij,n \ Ii,n | + |Ii,n \ Ij,n ||Ij,n \ Ii,n | − |Ii,n ∩ Ij,n |2 (1)

(2)

(1)

(2)

= |Ii,n ∩ Ij,n |2 + |Ii,n ||Ij,n |. To derive these identities we separated the increments of W (1) into increments over overlapping and non-overlapping intervals. In the case i = i , j =  j we obtain

(1) (2) (1) (2) (1) (2) (1) (2)   E Δi,n W (1) Δj,n W (1) Δi ,n W (1) Δj  ,n W (1) − |Ii,n ∩ Ij,n ||Ii ,n ∩ Ij  ,n |S

(1) (2) (2) (1) (2) (1) (2)   = E (Δi,n W (1) )2 Δj,n W (1) Δj  ,n W (1) − |Ii,n ∩ Ij,n ||Ii,n ∩ Ij  ,n |S

(2) (2) = E (Δi∩j W (1) + Δi∩j  W (1) + Δi\{j,j  } W (1) )2 Δj,n W (1) Δj  ,n W (1) (1) (2) (1) (2)   − |I ∩ I ||I ∩ I  |S i,n

=

(1) 2|Ii,n (1)

j,n

(2) (1) ∩ Ij,n ||Ii,n (2)

(1)

i,n

j ,n

(2) (1) ∩ Ij  ,n | − |Ii,n

(2)

(1)

(2)

∩ Ij,n ||Ii,n ∩ Ij  ,n |

(2)

= |Ii,n ∩ Ij,n ||Ii,n ∩ Ij  ,n | where Δi∩j W (1) , Δi∩j  W (1) , Δi\{j,j  } W (1) denote the increments of W (1) over (1)

(2)

(1)

(2)

(1)

(2)

(2)

the intervals Ii,n ∩ Ij,n , Ii,n ∩ Ij  ,n , Ii,n \ (Ij,n ∪ Ij  ,n ). By symmetry we obtain the same result in the case i =  i , j = j . In the case i =  i , j =  j we obtain using similar decompositions

(1) (2) (1) (2) (1) (2)   (2) (1) E Δi,n W (1) Δj,n W (1) Δi ,n W (1) Δj  ,n W (1) − |Ii,n ∩ Ij,n ||Ii ,n ∩ Ij  ,n |S 

× 1{I (1) ∩I (2) =∅} 1{I (1) i,n

(1)

(2)

i ,n

j,n

(1)

(2)

∩Ij  ,n =∅}

(2)

(1)

(2)

(1)

(2)

= |Ii,n ∩ Ij,n ||Ii ,n ∩ Ij  ,n | + |Ii,n ∩ Ij  ,n ||Ii ,n ∩ Ij,n | (1)

(2)

(1)

(2)

− |Ii,n ∩ Ij,n ||Ii ,n ∩ Ij  ,n | × 1{I (1) ∩I (2) =∅} 1{I (1) i,n

(1)

(2)

i ,n

j,n

(1)



(2)

∩Ij  ,n =∅}

(2)

= |Ii,n ∩ Ij  ,n ||Ii ,n ∩ Ij,n |1{I (1) ∩I (2) =∅} 1{I (1) i,n

j,n

i ,n

(2)

∩Ij  ,n =∅}

194

7 Estimating Quadratic Covariation (1)

(1)

which is always equal to zero because the intervals Ii,n , Ii ,n cannot at the same (2)

(2)

time both overlap with both other intervals Ij,n , Ij  ,n . Putting the results from those four cases together we obtain nρ2 E





 (1)

(2)

(1)

(1)

(2)

(1)

(2)

Δi,n W (1) Δj,n W (1) Δi ,n W (1) Δj  ,n W (1)

(2)

i,j:ti,n ∨tj,n ≤T i ,j  :ti ,n ∨tj  ,n ≤T (1)

(2)

(1)

(2)



− |Ii,n ∩ Ij,n ||Ii ,n ∩ Ij  ,n | 1{I (1) ∩I (2) =∅} 1{I (1) i,n

(2) ∩Ij  ,n =∅} i ,n

j,n

   S

2 n n = Gn 2,2 (T ) + ρ (H0,0,4 (T ) + H (T ))

which yields the claim. Proof of Theorem 7.3. The proof of (7.1) is very similar to the proof of (6.25) and of (6.31). It holds





l=1,2

(l) (3−l) i,j:ti,n ∨tj,n ≤T

V (f(1,1) , πn )T =

(1) (2) i,j:ti,n ∨tj,n ≤T



+



i,n

(l)

+

(2)

i,j:ti,n ∨tj,n ≤T

+



(1)

(2)

(3−l)

Δi,n C (l) Δj,n M (3−l) (q)1{I (l) ∩I (3−l) =∅} i,n

(7.17)

j,n

(l)

(3−l)

Δi,n (M (l) (q) + C (l) )Δj,n N (3−l) (q)1{I (l) ∩I (3−l) =∅} i,n

j,n

j,n

 (1)

(7.16)

j,n

j,n



l=1,2 i,j:t(l) ∨t(3−l) ≤T i,n

(7.15)

(2)

i,n

l=1,2 i,j:t(l) ∨t(3−l) ≤T

+

j,n

Δi,n M (1) (q)Δj,n M (2) (q)1{I (1) ∩I (2) =∅}





i,n

(1) (2) Δi,n B (1) (q)Δj,n B (2) (q)1{I (1) ∩I (2) =∅} i,n j,n (1)

(1) (2) i,j:ti,n ∨tj,n ≤T

+

(3−l)

(7.14)



−

(l)

Δi,n B (l) (q)Δj,n X (3−l) 1{I (l) ∩I (3−l) =∅}

i,j:ti,n ∨tj,n ≤T

(7.18) (1) (2) Δi,n C (1) Δj,n C (2) 1{I (1) ∩I (2) =∅} i,n j,n (1)

(7.19)

(2)

Δi,n N (1) (q)Δj,n N (2) (q)1{I (1) ∩I (2) =∅} . i,n

(7.20)

j,n

The S-conditional expectation of the absolute value of the term (7.14) is using similar arguments as for (6.26) bounded by Kq (|πn |T )1/2 T which vanishes as

7.4 The Proofs

195

n → ∞ for any q > 0. Further, the S-conditional expectation of the absolute value of the term (7.15) is using (1.8) bounded by

 (1)

(1)

(2)

Kq2 |Ii,n ||Ij,n |1{I (l) ∩I (3−l) =∅} ≤ 3Kq2 |πn |T T

(2)

i,n

i,j:ti,n ∨tj,n ≤T

j,n

which vanishes as well for n → ∞ and any q > 0. The term (7.16) can be treated similarly as (6.29) to find an upper bound for the S-conditional expectation of the square of (7.16) of the form K(eq )2 T 2 which vanishes as q → ∞. Further the expectation of the square of (7.17) is using (1.9), (1.10), (3.30) and Lemma 6.5 bounded uniformly in n by Keq T 2 which vanishes as q → ∞. The sum (7.18) vanishes as n → ∞ for any q > 0 because N (q) almost surely has only finitely many jumps and because of

 (l) i:ti,n ≤T

(l)

Δi,n (M (l) (q) + C (l) )1{S

(l) p ∈Ii,n }

→0

almost surely as n → ∞ for any jump time Sp of N (q). Finally we consider the terms (7.19) and (7.20) which make up the limit. First recall that

 (1)

(2)

(1)

P

(2)

Δi,n C (1) Δj,n C (2) 1{I (1) ∩I (2) =∅} −→ i,n

i,j:ti,n ∨tj,n ≤T

j,n



T 0

(1) (2)

ρs σs σs ds.

follows from Theorem 2.22 as bas been shown in Example 2.23. Further regarding (7.20) we observe

 (1)

(1)

(2)

(2)

Δi,n N (1) (q)Δj,n N (2) (q)1{I (1) ∩I (2) =∅}

i,j:ti,n ∨tj,n ≤T

i,n

j,n

P

−→



ΔN (1) (q)s ΔN (2) (q)s

s≤T

as N (q) almost surely only has finitely many jumps in [0, T ] which are separated by the observation scheme for |πn |T small enough. Combining the discussion of (7.14)–(7.20) above we obtain lim lim sup P(|V (f(1,1) , πn )T − [X (1) , X (2) ]T | > δ) = 0

q→∞ n→∞

for any δ > 0 which yields (7.1). The convergences in (7.2) follow from part b) of Theorem 2.38 respectively from part b) of Theorem 2.35.

196

7 Estimating Quadratic Covariation

Theorems 7.6 and 7.8 can be proven very similarly to Theorems 2 and 3 in [8]. The only difference is that we need a slightly more involved discretization of σ due to possible co-jumps of X and σ. This slightly more involved discretization has already been used in the proofs in Section 3.1. To illustrate the differences we lay out the rough structure of the proof of Theorem 7.6 but refer to [8] for the arguments which are similar. Proof of Theorem 7.6. Denote V AR,(l)

ΦT

=

 √  2 0

(l)

(l)

(σs )4 dG4 (s)

'



+2

T

(σ

(l) p:Sp ≤T

(l)

(l)

(l)

Sp −

1/2 U (l)

(l)

(l)

(l)

)2 δ− (Sp ) + (σSp )2 δ+ (Sp )US (l) ΔXS (l) p

(7.21)

p

and using the notation from the proof of Theorem 3.2 we define R(l) (n, q, r) = +

√

√  n

(l)

Δi,n C (l) (r) −



i:ti,n ≤T



n

 (l)

T 0

(σ (l) (r)s )2 ds

(l)

(l) (r, q). Δi,n N (l) (q)Δi,n C

i:ti,n ≤T V AR,(l)

ΦT

(q, r) =

 √  2

T 0

'



+2



(l)

(l)

(σ (l) (r)s )4 dG4 (s) (l)

1/2 U

(l)

(l)

(l)

(˜ σ (l) (r, q)S (l) − )2 δ− (Sq,p ) + (˜ σ (l) (r, q)S (l) )2 δ+ (Sq,p ) p

(l)

q,p

p:Sq,p ≤T

× US (l) ΔN (q)S (l) p

q,p

Following the structure of the proof of Theorem 3.2 we then show √ lim lim sup lim sup P(| n(V (l) (g2 , πn )T − [X (l) , X (l) ]T ) − R(l) (n, q, r)| > ε) = 0, q→∞

r→∞

n→∞

(7.22) (|ΦV AR,(l) − ΦV AR,(l) (q, r)| > ε) = 0, ∀ε > 0, lim lim sup P T T

q→∞

(7.23)

r→∞

L

s ΦT R(l) (n, q, r) −→

V AR,(l)

(q, r), ∀q, r > 0.

(7.24)

Here, (7.22) can be proven similarly as the corresponding statement in the proof of Theorem 2 in [8]. The statement (7.23) follows from P  [|ΦV AR,(l) − ΦV AR,(l) (q, r)|2 |X ] −→ E 0 T T

(7.25)

7.4 The Proofs

197

and Lemma 2.15. (7.25) can be shown as in Step 3 of the proof of Theorem 3.2 using the boundedness of σ and that the measures ΓV AR,(l) (·, dy) have uniformly bounded first moments. Further for proving (7.24), observe that the stable converV AR,(l) gence of the second term in R(l) (n, q, r) to the second term in ΦT (q, r) follows as in the proof of (3.25) while the convergence of the first term in R(l) (n, q, r) to V AR,(l) the first term in ΦT (q, r) and the asymptotic independence of the two terms (l) in R (n, q, r) is shown in the proof of Proposition 3 in [8]. Finally (7.4) follows from (7.22)–(7.24) and Lemma B.6. Proof of Theorem 7.8. Theorem 7.8 can be proven in the same way as Theorem 7.6 above where we use arguments from the proof of Theorem 3 in [8] and the proofs in Section 3.1 in the right places. As in Chapter 4 we first consider two propositions for the proofs of Theorem 7.11 and 7.13. V AR,(l)

Proposition 7.15. Suppose that Condition 7.10 is satisfied and let ΦT defined as in (7.21). Then it holds Mn         1  V AR,(l) ≤ ΥX  > ε → 0 P 1{Φ  V AR,(l) ≤Υ} − P ΦT

Mn

be

(7.26)

T ,n,m

m=1

for any X -measurable random variable Υ and all ε > 0. (l)

Proof. Denote by Sq,p , p ∈ N, an increasing sequence of stopping times which exhausts the jump times of N (l) (q) and define

% Y (P, n, m) = (V T,n

V AR,(l) 1/2

+2

)

P 

Δ

p=1



Un,m

(l) (l)

(l)

in (Sq,p ),n

(l)

× σ ˜n (t

X (l) 1{|Δ(l)

(l) (l) in (Sq,p ),n

(l)

(l)

(l)

(l)

in (Sq,p ),n (l)

+ ((˜ σn (t

, −))2 δˆn,m− (t

(l)

X (l) |>β|I

(l) (l)

(l)

in (Sq,p ),n (l)

(l) (l) in (Sq,p ),n

, +))2 δˆn,m,+ (t

(l) | } (l) (l) in (Sq,p ),n

)

(l) (l) (l) in (Sq,p ),n

)

1/2

× Un,i(l) (S (l) ),m 1{S (l) ≤T } , n

V AR,(l) 1/2

)

Y (P ) = (VT +2

P  p=1

q,p

q,p

U

' ΔXS (l)

q,p

(σ

(l) (l)

Sq,p −

(l)

(l)

(l)

(l)

(l)

)2 δ− (Sq,p ) + (σSp )2 δ+ (Sq,p )US (l) 1{S (l) ≤T } . q,p

q,p

198

7 Estimating Quadratic Covariation

1 % V AR,(l) , A1 = V V AR,(l) , Z%n,m Step 1. Then Lemma 4.9 with An,1 = V (s) = T,n T

Un,m , Z 1 (s) = U and



An,p+1 = Δ

(l) (l)

(l)

in (Sq,p ),n (l)

X (l) 1{|Δ(l)

(l) (l) in (Sq,p ),n

(l)

(l)

(l)

X (l) |>β|I



(l) | } (l) (l) in (Sq,p ),n

,

˜n (Sq,p , +) , σ ˜n (Sq,p , −), σ



Ap+1 = ΔX



(l) (l)

Sq,p

,σ

(l) (l)

Sq,p −

,σ



(l) (l)

Sq,p

,



(l) (l) p+1 %n,m Z (s) = δˆn,m,− (s), δˆn,m,+ (s), Un,i(l) (s),m ,



(l)

n



(l)

Z p+1 (s) = δ− (s), δ+ (s), Us , p = 1, . . . P , and the function ϕ defined such that (l)

ϕ((Ap , Z p (Sq,p−1 ))p=1,...,P +1 ) = Y (P ) (l)

p ϕ((An,p , Zn,m (Sq,p−1 ))p=1,...,P +1 ) = Y (P, n, m) (l)

(set Sq,0 = 0) yields

 P

Mn     1     Y (P ) ≤ ΥX  > ε → 0  1{Y (P,n,m)≤Υ} − P

Mn

(7.27)

m=1

as n → ∞ for any P ∈ N. Step 2. Next we prove lim lim sup

P →∞ n→∞

Mn   1   % V AR,(l)  > ε = 0 P Y (P, n, m) − Φ T,n,m Mn

(7.28)

m=1

for all ε > 0. Denote by Ω(P, q, n) the set on which there are at most P jumps of N (l) (q) in [0, T ] and two different jumps of N (l) (q) are further apart than |πn |T . Obviously, P(Ω(P, q, n)) → 1 for P, n → ∞ and any q > 0. On the set Ω(P, q, n) we have

 

% V AR,(l) |2 1Ω(P,q,n) S E |Y (P, n, m) − Φ T,n,m 

(l) ≤ E (Δi,n (X (l) − N (l) (q)))2 1{|Δ(l) X (l) |>β|I (l) | } (l)

(l)

i,n

(l)

ti,n ≤T,p:Sq,p ∈Ii,n



(l)

(l)

(l)

(l)

(l)

(l)

(l)

i,n

(l)



 

× (˜ σn (ti,n , −))2 δˆn,m,− (ti,n ) + (˜ σn (ti,n , +))2 δˆn,m,+ (ti,n ) (Un,i,m )2 S ≤



(l)

ti,n ≤T

(l) E (Δi,n (X (l) − N (l) (q)))2 1{|Δ(l) X (l) |>β|I (l) | } i,n

i,n

7.4 The Proofs ×

199

1 bn

 (l)



(l)

(l)

(l)

(l)

 

(l)

(Δj,n X (l) )2 (δˆn,m,− (ti,n ) + δˆn,m,+ (ti,n ))S

(l)

(l)

j =i:Ij,n ⊂(ti,n −bn ,ti,n +bn ]

(7.29) (l)

(l)

where the expectation of terms with Δi,n (X (l) − N (l) (q))Δj,n (X (l) − N (l) (q)), i = j vanishes because of (l)

(l)

E[Un,i,m Un,j,m |σ(F , {(δˆn,m,− (s), δˆn,m,+ (s))|s ∈ [0, T ]})] = E[Un,i,m Un,j,m ] = 0 for i =  j. The expression in the last line of (7.29) is using iterated expectations, Lemma 1.4 and inequality (2.77), compare Step 1 in the proof of Proposition 4.10 for details, bounded by (Kq |πn |T + (|πn |T )1/2− + eq )



Kn 

(l)

|Ii,n |

(l)

(l)

n|Ii+k,n |2

Kn 

(l)

|Ii+k ,n |

−1

k =−Kn

k=−Kn

ti,n ≤T



(l),n

≤ 2(Kq |πn |T + (|πn |T )1/2− + eq )G4

(T ).

Hence (7.28) follows from Condition 7.5(ii) and Lemma 2.15. Step 3. As in Step 3 in the proof of Proposition 4.10 we deduce

  P  V AR,(l)     Y (P ) ≤ ΥX −→  Φ P P ≤ ΥX T

(7.30)

for P → ∞ and as in that proof we obtain (7.26) from (7.27), (7.28) and (7.30). Proposition 7.16. Suppose that Condition 7.12 is satisfied and set ΦCOV = T (VTCOV )1/2 U + ZTCOV . Then it holds Mn     1 P 1{Φ  COV

Mn

m=1

 



T ,n,m ≤Υ}



 ΦCOV −P ≤ Υ X  > ε → 0 T

(7.31)

for any X -measurable random variable Υ and all ε > 0. Proof. The structure of this proof is identical to the structure of the proof of Proposition 7.15. We denote by Sq,p , p ∈ N, an increasing sequence of stopping times which exhausts the jump times of N (q) and define COV 1/2 %T,n ) Un,m Y (P, n, m) = (V

+

P 



1{Sq,p ≤T }

(1) (2) i,j:ti,n ∨tj,n ≤T

p=1



1{i=i(1) (S n

q,p )

(2)

or j=in (Sq,p )}

(1)

× 1{I (1) ∩I (2) =∅} Δi,n X (1) 1{|Δ(1) X (1) |>β|I (1) | } i,n

j,n

i,n

i,n

200

7 Estimating Quadratic Covariation



(2)

(1)

n × σ n (ti,n , −)˜ ρn (τi,j , −) (2)

$

(1),−

%n,m (τ n )U L i,j n,(i,j),m

(1)

n + σ n (ti,n , +)˜ ρn (τi,j , +)



(2)

$

(1),+

% n,m (τ n )U R i,j n,(i,j),m

(1)

n n %n,m (τi,j + (σ n (ti,n , −))2 (1 − (˜ ρn (τi,j , −))2 )L ) (2)

(1)

n n % n,m (τi,j + (σ n (ti,n , +))2 (1 − (˜ ρn (τi,j , +))2 )R )

$ +

(2)

(1)

(2)

(2)

(1)



(2)

j,n

(1)

(2)

(1)

(2)

(2)

(1)

$

(4)



(1),−

%n,m (τ n )U L i,j n,(i,j),m

j,n

$

(1),+

% n,m (τ n )U R i,j n,(i,j),m

+ σ n (tj,n , +)

$

(3)

Un,(i,j),m

%n,m (τ n ) + (σ n (t , +))2 R % n,m (τ n )U (σ n (ti,n , −))2 L i,n i,j i,j n,(i,j),m

+ Δj,n X (2) 1{|Δ(2) X (2) |>β|I (2) | } σ n (tj,n , −)

+

1/2

(2)

(1)

(1)

(2)

(1)

(2)

%n,m (τ n ) + (σ n (t , +))2 R % n,m (τ n )U (σ n (tj,n , −))2 L j,n i,j i,j n,(i,j),m

× 1{|Δ(1) X (1) |=max i,n

k:I

× 1{|Δ(2) X (2) |=max j,n

(1) (2) ∩I

=∅ j,n k,n

k:I

(2) (1) ∩I

=∅ i,n k,n



(1)

|Δk,n X (1) |} (2)

|Δk,n X (2) |}

.

Further we define Y (P ) similarly as ΦCOV with the only difference that in the T term ZTCOV we do not sum over all jump times Sp ≤ T , but only over the jump times Sq,p ≤ T . Suppose that at Sq,p there is a common jump of X (1) and (1) X (2) . Then for |πn |T small enough we have that Δ (1) X (1) is the largest (1)

in (Sq,p ),n

increment among all increments over intervals Ik,n which overlap with I and Δ (2) Ik,n

(2) (2)

in (Sq,p ),n

X

(2)

(2)

is the largest increment among all increments over intervals

which overlap with I (2)

(2) in (Sq,p ),n

(1)

(1)

(1)

in (Sq,p ),n

. Hence exactly for in p = in (Sq,p ) and

jpn = in (Sq,p ) we obtain a summand which is asymptotically not equal to zero. Next suppose that only X (1) jumps at Sq,p . As argued above only summands (1) with i = in (Sq,p ) will not vanish in the limit. Further for exactly one j with (1) (2) (2) Ij,n ∩ I (1) = ∅ we have |Δj,n X (2) | = maxk:I (2) ∩I (1) =∅ |Δk,n X (2) | because in (Sq,p ),n

k,n

i,n

(2)

σ (l) does not vanish by Condition 7.12 and the size of this |Δj,n X (2) | vanishes as (2)

n → ∞ because X does not jump at Sq,p . Furthermore by construction the F-conditional distribution of (1)

Δin ,n X (1) 1{|Δ(1) p

×



in p ,n

(1)

X (1) |>β|Iin ,n | } p

(2) (1) σ n (tin ,n , −)˜ ρn (τinnp ,j , −) p

$

(1),−

%n,m (τ nn )U n L i ,j n,(i ,j),m p

p

7.4 The Proofs

201

(2)

(1)

+ σ n (tin ,n , +)˜ ρn (τinnp ,j , +)

 +

$

(1),+

% n,m (τ nn )U n R i ,j n,(i ,j),m

p

p

p

(2) (1) %n,m (τinn ,j ) (σ n (tin ,n , −))2 (1 − (˜ ρn (τinnp ,j , −))2 )L p p

1/2 (2) (1) (3) % n,m (τinn ,j ) + (σ n (tin ,n , +))2 (1 − (˜ ρn (τinnp ,j , +))2 )R Un,(in ,j),m p p p

'

(2)

(1)

(2)

(2)

(1)

(2)

(4)

%n,m (τ nn ) + (σ n (t n , +))2 R % n,m (τ nn )U n (σ n (tin ,n , −))2 L i ,n i ,j i ,j n,(i ,j),m

+

p

p

(2) + Δj,n X (2) 1{|Δ(2) X (2) |>β|I (2) | } j,n j,n (1)

$

(2)

+ σ n (tj,n , +)



p

(1) (2) σ n (tj,n , −)

p

%n,m (τ nn )U (1),− L ip ,j n,(in p ,j),m

(1),+

% n,m (τ nn )U n R i ,j n,(i ,j),m p

'

p

$

p

(1) (2) (1) (2) n 2 % (1) (τ n )U (2) %(1) (σ n (tj,n , −))2 L n,m (τin ,j ) + (σ n (tj,n , +)) R n,m in ,j n,(in ,j),m

+



p

p



p

(1)

(2)

asymptotically does not depend on j for all j ∈ N with Iin ,n ∩ Ij,n = ∅ because p

(1)

j only has an influence on the correlation between the term involving Δin ,n X (1) p

(2)

and the term involving Δj,n X (2) which becomes irrelevant as the second term vanishes anyways. Hence based on these considerations and similar arguments for idiosyncratic jumps in X (2) we obtain that Y (P, n, m) is asymptotically equivalent to COV 1/2 %T,n Y (P, n, m) = (V ) Un,m

+

P 

(1) (2) i,j:ti,n ∨tj,n ≤T

p=1





1{Sq,p ≤T }

1{i=i(1) (S n

q,p )

(2)

and j=in (Sq,p )}

(1)

× Δi,n X (1) 1{|Δ(1) X (1) |>β|I (1) | } ×



i,n

(2) (1) n σ n (ti,n , −)˜ ρn (τi,j , −) (2)

$

(1)

i,n

(1),−

%n,m (τ n )U L i,j n,(i,j),m

n + σ n (ti,n , +)˜ ρn (τi,j , +)



(2)

$

(1),+

% n,m (τ n )U R i,j n,(i,j),m

(1)

n n %n,m (τi,j + (σ n (ti,n , −))2 (1 − (˜ ρn (τi,j , −))2 )L ) (2)

(1)

n n % n,m (τi,j + (σ n (ti,n , +))2 (1 − (˜ ρn (τi,j , +))2 )R )

$ +

(2)

(1)

(2)

(2)

(1)

(1)

(2)

1/2

(2)

(3)

Un,(i,j),m (4)

%n,m (τ n ) + (σ n (t , +))2 R % n,m (τ n )U (σ n (ti,n , −))2 L i,n i,j i,j n,(i,j),m 

(2)

+ Δj,n X (2) 1{|Δ(2) X (2) |>β|I (2) | } σ n (tj,n , −) j,n

(1)

(2)

+ σ n (tj,n , +)

$

j,n

(1),+

% n,m (τ n )U R i,j n,(i,j),m

$

(1),−

%n,m (τ n )U L i,j n,(i,j),m



202

7 Estimating Quadratic Covariation

$ +

(1)

(2)

(1)

(1)

(2)

(1)

(2)

%n,m (τ n ) + (σ n (t , +))2 R % n,m (τ n )U (σ n (tj,n , −))2 L j,n i,j i,j n,(i,j),m



in the sense that they have for m = 1, . . . , Mn asymptotically identical common F-conditional distributions. (l) Step 1. Note that it holds σ (3−l) (t (l) , −) = σ ˜ (3−l) (Sq,p , −)+oP (1). Hence, in (Sq,p ),

using Lemma 4.9, Corollary 6.4 and Condition 7.12 we obtain similarly as in Step 2 of the proof of Proposition

 P

Mn     1     Y (P ) ≤ ΥX  > ε → 0  1{Y  (P,n,m)≤Υ} − P

Mn

m=1

as n → ∞ for any P ∈ N which then yields

 P

Mn     1     Y (P ) ≤ ΥX  > ε → 0  1{Y (P,n,m)≤Υ} − P

Mn

(7.32)

m=1

based on the above considerations. Step 2. Next we prove lim lim sup

P →∞ n→∞

Mn   1    % COV P Y (P, n, m) − Φ T,n,m > ε = 0 Mn

(7.33)

m=1

for all ε > 0. Denote by Ω(P, q, n) the set on which there are at most P jumps of N (q) in [0, T ] and two different jumps of N (q) are further apart than |πn |T . Obviously, P(Ω(P, q, n)) → 1 for P, n → ∞ and any q > 0. On the set Ω(P, q, n) we obtain similarly as in Step 2 of the proof of Proposition 7.16 the following estimate

 

2  % COV E |Y (P, n, m) − Φ T,n,m | 1Ω(P,q,n) F   (l) ≤K

l=1,2

×

1

(l) (3−l) il ,i3−l :ti ,n ∨ti ≤T l 3−l ,n

(Δil ,n (X (l) − N (l) (q)))2 1{|Δ(l)

il ,n X



bn

(3−l)

k:Ik,n

× 1{I (l) il

(l) |>β|I (l) | } il ,n

(3−l)

(Δk,n X (3−l) )2



(l) (l) (l) (l) −bn ,ti ,n −|πn |T )∪(ti ,n ,ti ,n +bn ] l ,n l l l

⊂(ti

(3−l)

=∅} 3−l ,n

,n ∩Ii



l=1,2

1{|Δ(l)

il ,n X

(l) |=max (l) (3−l) k:I ∩I

=∅ i3−l ,n k,n

n %COV % COV n × E [L n,m (τil ,i3−l ) + Rn,m (τil ,i3−l )

%n,m +L

COV,(3−l)

% n,m (τinl ,i3−l ) + R

COV,(3−l)

(l)

|Δk,n X (l) |}

 

(τinl ,i3−l )F .

(7.34)

7.4 The Proofs

203

COV %n,m From the definition of Z (s) in (7.11) we obtain n % COV n E[L%COV n,m (τil ,i3−l ) + Rn,m (τil ,i3−l )

%n,m +L

COV,(3−l)

≤

1 2bn



il ,i3−l :|τin ,i

3−l

% n,m (τinl ,i3−l ) + R

COV,(3−l)

(3−l)

Mn

n −τi,i

3−l

(τin ,i l

|≤bn

 

(τinl ,i3−l )F

(1)

3−l

(2)

)|Ii ,n ∩ Ii

3−l ,n

| + n|πn |T

|πn |T 2bn (7.35)

where



(l)

Mn (s) = n

(l)

(l) i:ti,n ≤T

|Ii,n |1{I (l) ∩I (3−l) i,n

(3−l) in (s),n

=∅}

, l = 1, 2.

Using the estimate (7.35) we find that we can split the discussion of (7.34) into two parts. First we discuss (7.34) with the F -conditional expectation replaced by |πn |T /(2bn ). This expression is bounded by Kn(|πn |T )2  2bn ×

1 bn



(l)

(Δi,n (X (l) − N (l) (q)))2

l=1,2 i:t(l) ≤T i,n



(3−l)

(Δk,n X (3−l) )2

 (7.36)

(3−l) (l) (l) (l) (l) k:Ik,n ⊂(ti,n −bn ,ti,n −|πn |T )∪(ti,n ,ti,n +bn ]

(l)

because each increment (Δi,n (X (l) − N (l) (q)))2 occurs in the sum at most once. P

The expression (7.36) vanishes as n → ∞ due to |πn |T /bn −→ 0 by Condition 7.12 because the S-conditional expectation of the sum in (7.36) is bounded in probability using iterated expectations and Lemma 1.4. Note to this end that by (3−l) (l) (l) (l) (l) construction no interval Ik,n ⊂ (ti,n − bn , ti,n − |πn |T ) ∪ (ti,n , ti,n + bn ] overlaps (l)

with Ii,n . Next, we discuss (7.34) where we replace the expectation by the sum over the (3−l) n Mn (τi ,i ) from (7.35). By an index change as in (4.33) this expression is l 3−l equal to K





l=1,2

(l) (3−l) il ,i3−l :ti ,n ∨ti ≤T l 3−l ,n

×

1 2bn

(3−l)

Mn



il ,i3−l :|τin ,i

3−l

n −τi,i

(1)

(2)

(τinl ,i3−l )|Iil ,n ∩ Ii3−l ,n | (l)

3−l

|≤bn

(Δi ,n (X (l) − N (l) (q)))2 1{|Δ(l) l

i ,n l

(l)

X (l) |>β|Ii ,n | } l

204 ×

7 Estimating Quadratic Covariation

1



bn

(3−l)

k:Ik,n

× 1{I (l)

i ,n l



×

×

×

(l) (l) (l) (l) −bn ,ti ,n −|πn |T )∪(ti ,n ,ti ,n +bn ] l ,n l l l

⊂(ti

(3−l)

=∅} ,n 3−l

1{|Δ(l)

i ,n l



l=1,2

X (l) |=max

k:I



(l) (3−l) ∩I 

=∅ k,n i ,n 3−l

(3−l)

Mn

(l)

(l) |≤bn +2|πn |T l ,n

l

bn

(1)

(2)

(τinl ,i3−l )|Iil ,n ∩ Ii3−l ,n |

(l)

il :|ti ,n −ti

1

(l)

|Δk,n X (l) |}

(l) (3−l) il ,i3−l :ti ,n ∨ti ≤T l 3−l ,n



1 2bn



∩Ii

l=1,2

≤K

(3−l)

(Δk,n X (3−l) )2

(3−l)

k:Ik,n

(Δi ,n (X (l) − N (l) (q)))2 1{|Δ(l)

i ,n l

l



(3−l)

(Δk,n X (3−l) )2

(l)

(l)

(l)

(l)

l

l

l

l

(l)

X (l) |>β|Ii ,n | } l



⊂(ti ,n −bn ,ti ,n −|πn |T )∪(ti ,n ,ti ,n +bn ]

× 1{I (l)

(7.37)

(3−l) ∩Ii ,n =∅} i ,n l 3−l

(l)

where we again used that each increment (Δi ,n (X (l) − N (l) (q)))2 occurs in the l

sum at most once. Using iterated expectations, Lemma 1.4 and inequality (2.77) similarly as for (7.36) we obtain that the S-conditional expectation of (7.37) is bounded by K





l=1,2

(l) (3−l) il ,i3−l :ti ,n ∨ti ≤T l 3−l ,n

×

1 2bn



(l)

×



(l)

(Kq |πn |T + (|πn |T )1/2− + eq )|Ii ,n | l

(l) |≤bn +|πn |T l ,n



n

 (3−l)

j:tj,n

≤K

(3−l)

bn + |πn |T (Kq |πn |T + (|πn |T )1/2− + eq ) bn l=1,2

×

(l)

(τinl ,i3−l )|Iil ,n ∩ Ii3−l ,n |

il :|ti ,n −ti l

≤K

(3−l)

Mn

(l)

(3−l)

|Iil ,n ∩ Ii3−l ,n |

(l) (3−l) ∨ti ≤T l ,n 3−l ,n

il ,i3−l :ti

|Ij,n |(3−l) |1{I (3−l) ∩I (l) ≤T

 b + |π | 2 n n T bn

j,n

il ,n =∅}

(Kq |πn |T + (|πn |T )1/2− + eq )Gn 2,2 (T )

2bn bn

7.4 The Proofs

205

which vanishes as first n → ∞ and then q → ∞. Hence, alltogether we have shown that the S-conditional expectation of (7.34) vanishes as n → ∞ and then q → ∞ which yields (7.33) using Lemma 2.15. Step 3. As in Step 3 in the proof of Proposition 4.10 we deduce

  P  COV     Y (P ) ≤ ΥX −→  ΦT P P ≤ Υ X

(7.38)

for P → ∞ and as in that proof we obtain (7.31) from (7.32), (7.33) and (7.38). Proof of Theorems 7.11 and 7.13. Analogously as in the proof of Theorem 4.3, Propositions 7.15 and 7.16 yield that under Conditions 7.10 respectively 7.12 it holds

(√n([X (l) , X (l) ]T − V (l) (g2 , πn )T ) ≤ Q %V AR,(l) (α)|F) = α, lim P T,n √ (1) (2) ( n([X , X ]T − V (f(1,1) , πn )T ) ≤ Q %COV lim P T,n (α)|F) = α

n→∞ n→∞

for any α ∈ [0, 1]. These convergences together with the fact that the F -conditional V AR,(l) distributions of ΦT and ΦCOV are symmetrical yield (7.10) and (7.12). T

8 Testing for the Presence of Jumps When choosing a suitable continuous time process e.g. to model an economic or financial time series one of the first steps is to decide whether a stochastic model is sufficient that produces continuous paths or whether jumps have to be incorporated. To this end, one is faced with the problem to infer from observed data (which is usually only available at discrete time points) whether the underlying model is continuous or allows for jumps. Sometimes this problem is relatively simple to solve e.g. in the situation when very large jumps are easy to identify in a visualization of the time series data. However, when only small but very frequent jumps are present the data might look very similar to observations of continuous time processes. Further, the method of visual inspection becomes infeasible when working with very large amounts of data. Hence automated methods and ideally mathematically precise tests are needed to tackle this problem. The decision whether to use models with or without jumps is not only of theoretical interest as to which model best fits reality but is also of great practical relevance since models with and without ” jumps do have quite different mathematical properties and financial consequences (for option hedging, portfolio optimization, etc.)”([2], page 184, second paragraph). Next, we formalize the problem discussed above under the assumption that the data is generated by discrete time observations of an Itˆ o semimartingale X (l) as (l) in (1.1) at random and irregular times ti,n , i ∈ N0 . Here, we are working within (l)

a high-frequency setting, where we observe one realized path Xt (ω), t ∈ [0, T ], up to a fixed time horizon T > 0 and consider asymptotics as the mesh of the observation times tends to zero. Hence as we are only considering one specific realization it is impossible to decide whether the underlying model in principle allows for jumps or not. This is due to the fact that the model might allow for jumps but no jumps have been realized for the ω which we observe. However, (l) we are able to decide asymptotically whether jumps in the realized path Xt (ω), t ∈ [0, T ], are present or not we are observing this path at finer and finer grids. Hence in this chapter, we will not construct statistical tests that allow to decide whether the underlying model allows for jumps but we will construct a statistical test which allows to decide whether jumps are present in a realized path or not. (l) So mathematically we are looking for a test based on the observations X (l) (ω), ti,n

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 O. Martin, High-Frequency Statistics with Asynchronous and Irregular Data, Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics, https://doi.org/10.1007/978-3-658-28418-3_8

208

8 Testing for the Presence of Jumps

i ∈ N0 , which allows to decide to which of the following two subsets of Ω J,(l)

ΩT

C,(l) ΩT J,(l)

ω belongs. Here, ΩT

(l)

= {∃t ∈ [0, T ] : ΔXt =

J,(l) (ΩT )c

=

(l) {ΔXt

= 0}, = 0 ∀t ∈ [0, T ]}

is the set of all ω for which the path of X up to T has at

C,(l) ΩT

least one jump and is the set of all ω for which the path of X is continuous on [0, T ]. The test constructed in the following section can be understood as a generalization of tests described in [2], in Section 11.4 of [30] or in Chapter 10 of [3] which are based on equidistant and deterministic observation times to the setting of irregular and asynchronous observation times.

8.1 Theoretical Results J,(l)

To decide whether ω ∈ ΩT

J,(l)

behave differently on ΩT

C,(l)

or ω ∈ ΩT C,(l)

and ΩT

is true we need to find statistics that

. Recall that by Corollary 2.2 it holds P

V (l) (g p , πn )T −→ B (l) (g p )T





for g p (x) = |x|p , p > 2, where B (l) (g p )T (ω) = C,(l)

to zero if and only if ω ∈ ΩT positive value on

J,(l) ΩT



(l)

s≤T

|ΔXs (ω)|p is equal

. Hence V (l) (g p , πn )T converges to a strictly C,(l)

and to zero on ΩT

. Then based on the asymptotics of

V (l) (g p , πn )T it is in principle possible to decide whether ω ∈ ΩT

J,(l)

C,(l)

or ω ∈ ΩT

.

In Chapter 3 we found that a central limit theorem for V (l) (g p , πn )T can only be derived for p > 3 and for mathematical reasons it turns out that it is especially convenient to work with p = 4 in which case we have g 4 (x) = g4 (x) = x4 . Further, as we are going to construct a statistical test under the null hypothesis J,(l) that jumps do exist, i.e. ω ∈ ΩT , it is advantegeous to work with a statistic that J,(l)

and not to an arbitrary limit greater than zero. converges to a fixed value on ΩT To this end we define the statistic J,(l)

Ψk,T,n =

V (l) (g4 , [k], πn )T , kV (l) (g4 , πn )T

k ≥ 2; compare (2.88) for the definition of V (l) (g4 , [k], πn )T . Here, the func(l) (l) tional V (l) (g4 , [k], πn )T relies on increments of X (l) over the intervals (ti−k,n , ti,n ]

while V (l) (g4 , πn )T relies on increments over the original observation intervals (l) (l) (ti−1,n , ti,n ]. Hence by evaluating the ratio of the two we are comparing the

8.1 Theoretical Results

209

expression V (l) (g4 , πn )T based on data sampled at different frequencies, once at the original frequency and once at 1/k times the original frequency. As the increments of continuous Itˆ o semimartingales and Itˆ o semimartingales with jumps scale differently with the length of the observation interval, compare the discussion J,(l) at the beginning of Section 2.2, we will see that the limit of Φk,T,n is different on J,(l)

ΩT

C,(l)

and on ΩT

. (l)

Remark 8.1. In the setting of equidistant observation times ti,n = i/n our statistic becomes

nT 

J,(l) Ψk,T,n

=

(l) (l) i=k g(Δi,k,n X ) . T /n (l) k i=1 g(Δi,n X (l) )

(8.1)

On the contrary in [2] a test is constructed based on the statistic

 J,(l) = Ψ k,T,n

nT /k

(l) g(Δik,k,n X (l) ) i=1 T /n (l) g(Δi,n X (l) ) i=1

(8.2)

where at the lower observation frequency n/k only increments over certain observation intervals Iik,k,n enter the estimation. Intuitively it seems that using the statistic (8.1) should be better than using (8.2), because in (8.1) we utilize the available data more exhaustively by using all increments at the lower observation frequency. This intuition is confirmed by Proposition 10.19 in [3] where central limit theorems are developed for both (8.1) and (8.2) and it is shown that (8.1) has a smaller asymptotic variance.  J,(l)

To analyse the limit of Ψk,T,n recall that Corollary 2.2 and Theorem 2.40(a) yield P

P

V (l) (g4 , πn )T −→ B (l) (g4 )T respectively V (l) (g4 , [k], πn )T −→ kB (l) (g4 )T . Hence J,(l) J,(l) C,(l) on ΩT the statistic Ψk,T,n converges to 1. On ΩT on the other hand we J,(l)

obtain that both numerator and denominator in the definition of Ψk,T,n converge to zero. However, we obtain P

nV (l) (g4 , πn )T 1ΩC,(l) −→ 3 T



T 0

(l)

(l)

(σs )4 dG4 (s)1ΩC,(l)

n (l) P V (g4 , [k], πn )T 1ΩC,(l) −→ 3 k2 T

T



T 0

(8.3)

(l) [k],(l) (σs )4 dG4 (s)1ΩC,(l) T

(l),n

by Corollary 2.19 and Theorem 2.41(a) if we assume that the functions Gp [k],(l),n Gp

(l) Gp

[k],(l) Gp

and

converge to some functions respectively as n → ∞. In the following condition we state precise assumptions under which the above considerations are valid.

210

8 Testing for the Presence of Jumps

Condition 8.2. Let Condition 1.3 be fulfilled. Further let (πn )n∈N be exogenous, T let the process σ fulfil 0 |σs |ds > 0 almost surely and suppose that the func(l),n

tions G4

[k],(l),n

(t), G4

(t) converge pointwise on [0, T ] in probability to strictly

(l)

[k],(l)

increasing functions G4 , G4 J,(l)

: [0, T ] → [0, ∞). J,(l)

The convergence of Ψk,T,n on ΩT already follows under Condition 1.3. The remaining assumptions in Condition 8.2 are only needed to prove the convergence J,(l) C,(l) (l),n [k],(l),n of Ψk,T,n on ΩT . The exogeneity and the convergence of G4 (t), G4 (t) are required such that (8.3) holds while

T

(l)

|σs |ds > 0 and that the function G4

0

is strictly increasing is needed such that the limit of nV (l) (g4 , [k], πn )T is never equal to zero. C,(l) J,(l) we denote To shorten the notation for the limit of Ψk,T,n on ΩT J,(l)

CT



T

= 0

(l)

(l)

(σs )4 dG4 (s),



J,(l)

Ck,T =

T 0

(l)

[k],(l)

(σs )4 dG4

(s).

Theorem 8.3. Under Condition 8.2 it holds J,(l) Ψk,T,n

J,(l)

⎧ ⎨1,

P

−→

J,(l)

⎩

J,(l)

J,(l) kCk,T J,(l) CT

on ΩT ,

on

, (8.4)

C,(l) ΩT .

C,(l)

Here, the limit kCk,T /CT on ΩT is always different from 1 if the observation scheme is not too degenerated. This property is illustrated in the following remark. Remark 8.4. We obtain (l),n

G4

(l),n

(t) − G4

[k],(l),n

(s) + O(|πn |t ) ≤ kG4

[k],(l),n

(t) − kG4

(l),n

(s) ≤ kG4

(l),n

(t) − kG4

(s)

for all t ≥ s ≥ 0 from the series of elementary inequalities k 

a2i ≤

k 

i=1

2 ai

≤k

i=1

k 

a2i

(8.5)

i=1

which holds for any a1 , . . . , ak ≥ 0, k ∈ N. Here, the second inequality follows from the Cauchy-Schwarz inequality. Equality in (8.5) holds for a1 ≥ 0, (l),n a2 = . . . = ak = 0 respectively a1 = . . . = ak > 0. The relations of G4 and [k],(l),n

G4

are preserved in the limit which yields (l)

(l)

[k],(l)

G4 (t) − G4 (s) ≤ kG4

[k],(l)

(t) − kG4

J,(l)

J,(l)

for all t ≥ s ≥ 0. Hence we get kCk,T /CT

(l)

(l)

(s) ≤ kG4 (t) − kG4 (s)

∈ [1, k].



8.1 Theoretical Results

211 P

J,(l)

J,(l)

J,(l)

Based on the fact that Ψk,T,n −→ 1 on ΩT

C,(l)

and that Ψk,T,n converges on ΩT J,(l)

J,(l)

to a random variable which is strictly greater than 1 if and only if kCk,T > CT by Remark 8.4, we will later construct a test with critical region Ck,T,n = {Ψk,T,n > 1 + ck,T,n } J,(l)

J,(l)

J,(l)

(8.6)

for an appropriate series of decreasing random positive numbers ck,T,n , n ∈ N. J,(l)

Next, we illustrate the result from Theorem 8.3 by looking at two prominent observation schemes: first Poisson sampling, compare Definition 5.1, which is truly random and irregular and second equidistant sampling for which results exist in the literature and which therefore serves as a kind of benchmark. Example 8.5. In the case of Poisson sampling Condition 8.2(ii) is fulfilled. To (l) this end note that n|Ii,k,n | is Gamma-distributed with parameters k and λl . Hence (l)

(l)

E[(n|Ii,k,n |)2 ] = k/λl , E[(n|Ii,k,n |)2 ] = k(k + 1)/λl and by Lemma 5.4 we obtain (l)

G4 (t) =

2 t, λ

[k],(l)

G4

(t) =

k+1 t. kλ



This yields that the limit under the alternative is equal to (k + 1)/2. (l)

Remark 8.6. In the case of equidistant synchronous observations, i.e. ti,n = i/n, (l)

[k],(l)

it holds G4 (t) = G4

(t) = t which yields J,(l)

kCk,T

J,(l)

= k.

CT J,(l)

C,(l)

Hence in this setting Ψk,T,n converges on ΩT

to a known deterministic limit J,(l)

as well which also allows to construct a test using Ψk,T,n for the null hypothesis of no jumps; compare Section 10.3 in [3]. This is not immediately possible in the irregular setting, unless the law of the generating mechanism is known to the statistician.  J,(l)

As a next step we develop a central limit theorem for the statistic Ψk,T,n under the null hypothesis that jumps are present i.e. we derive a central limit theorem J,(l) which holds on ΩT . To motivate the structure of the upcoming central limit theorem we consider as in Section 7.2 a toy example of the form toy,(l)

Xt

(l)

= σ (l) Wt

+

 s≤t

ΔN (l) (q)s .

212

8 Testing for the Presence of Jumps

First note that it holds √

√

J,(l)

n(Ψk,T,n − 1) =

n(V (l) (g4 , [k], πn )T − kV (l) (g4 , πn )T ) kV (l) (g4 , πn )T

(8.7)

and therefore it remains to consider the nominator as the denominator converges J,(l) (l) on ΩT to B (l) (g4 )T > 0. Denote by Sq,p , p ∈ N, an enumeration of the jump (l) times of N (q). In our toy example we then have on the set where any two jumps of N (l) (q) are further apart than k|πn |T √

n(V (l) (g4 , [k], πn )T − kV (l) (g4 , πn )T ) 4 √    (l) (l) = n Δi−k,n X toy,(l) + . . . + Δi,n X toy,(l) (l)

i:ti,n ≤T



−k



(l)

Δi,n X toy,(l)

4 

(l)

i:ti,n ≤T

=

√  n



4(ΔX

(l)

p:Sq,p ≤T

toy,(l) 3 ) (l) Sq,p

k−1 

σ (l) Δ

j=0



−k

 (l)

p:Sq,p ≤T

(ΔX

toy,(l) 3 √ ) n (l) Sq,p

(l)

(l)

in (Sq,p )+j,k,n

4(ΔX

(l) p:Sq,p ≤T

=4

(l)

W (l)

toy,(l) 3 (l) (l) ) σ Δ (l) (l) W (l) (l) Sq,p in (Sq,p ),n

k−1 

|k − j|σ (l) Δ

j=−(k−1),j =0

(l) (l)

(l)

in (Sq,p )+j,n

 + oP (1)

W (l) + oP (1)

under suitable conditions. Here, the oP (1)-term contains all sums over terms in (l) which higher powers of the increments Δi,n W (l) occur that originate from the expansion of



(l)

Δi,k ,n X toy,(l)

4



(l)

(l)

4

= σ (l) Δi,k ,n W (l) + Δi,k ,n N (l) (q) , k = 1, k.

A detailed discussion of the asymptotically negligible terms will be given in the proofs section. Here, the S-conditional variance of the term √

n

k−1  j=−(k−1),j =0

|k − j|Δ

(l) (l)

(l)

in (Sq,p )+j,n

W (l)

8.1 Theoretical Results

213

|I |I |I

(l) (l)

in (s),3,n

(l) (l)

in (s)+1,3,n

(l) (l)

in (s)+2,3,n

|

|

| s

t

(l) (l)

in (s)−3,n

t

(l) (l)

in (s)−2,n

t

(l)

t

(l)

in (s)−1,n

3|I

(l)

(l) (l)

(l)

in (s),n

in (s),1,n

t

(l)

t

(l)

in (s)+1,n

(l) (l)

in (s)+2,n

|

(l)

(l)

Figure 8.1: Illustrating the origin of ξk,n,− (s), ξk,n,+ (s) for k = 3. (l)

(l)

is equal to ξk,n,− + ξk,n,+ where (l)

ξk,n,− (s) = n

k−1 

(k − j)2 |I

j=1 (l)

ξk,n,+ (s) = n

k−1 

(k − j)2 |I

j=1

(l) (l)

in (s)−j,1,n (l) (l)

in (s)+j,1,n

|,

|.

The form of these terms is illustrated in Figure 8.1. To obtain a central limit theorem we then need to assume that the random vari(l) (l) ables ξk,n,− (s), ξk,n,+ (s) converge in a suitable sense as n → ∞. All assumptions J,(l)

which are required to obtain a central limit theorem for Ψk,T,n are summarized in the following condition. Condition 8.7. The process X (l) and the sequence of observation schemes (πn )n∈N fulfil Condition 8.2. Further the following additional assumptions on the observation schemes hold: (i) We have |πn |T = oP (n−1/2 ). (ii) The integral

 [0,T ]P

g(x1 , . . . , xP )E

P  p=1



(l)

(l)



hp ξk,n,− (xp ), ξk,n,+ (xp ) dx1 . . . dxP

214

8 Testing for the Presence of Jumps converges for n → ∞ to

 [0,T ]P

P  

g(x1 , . . . , xP )

p=1

R

hp (y) ΓJ,(l) (xp , dy)dx1 . . . dxP

for all bounded continuous functions g : RP → R, hp : R2 → R, p = 1, . . . , P , and any P ∈ N. Here ΓJ,(l) (·, dy) is a family of probability measures on [0, T ] T with uniformly bounded first moments and 0 ΓJ,(l) (x, {(0, 0)})dx = 0. Part (i) of Condition 8.7 guarantees that |πn |T vanishes sufficiently fast which is required in the proof, while part (ii) of 8.7 yields that the random variables (l) (l) (ξk,n,− (s), ξk,n,+ (s)) converge in law in a suitable sense. To describe the limit in the upcoming central limit theorem we define



J,(l)

Φk,T = 4

(ΔX

(l) p:Sp ≤T

(l)

3 (l) )

'

(l)

(l)

(l)

(l)

(σS (l) − )2 ξk,− (Sp ) + (σS (l) )2 ξk,+ (Sp )U

Sp

p

(l) (l)

Sp

p

. (8.8)

(l)

(l)

Here, (Sp )p≥0 denotes an enumeration of the jump times of X (l) and the ξk,− (s), (l)

ξk,+ (s), s ∈ [0, T ], are independent random variables which are distributed accord(l)

ing to ΓJ,(l) (s, dy) and the Us , s ∈ [0, T ], are i.i.d. standard normal distributed (l) (l) (l) random variables. Both the (ξk,− (s), ξk,+ (s)) and the Us are independent of

).  F, P X (l) and its components and defined on an extended probability space (Ω, J,(l) Note that Φk,T is well-defined because the sum in (8.8) is almost surely absolutely (l)

convergent and independent of the choice for the enumeration Sp , p ∈ N; compare Proposition 4.1.3 in [30]. Theorem 8.8. If Condition 8.7 holds, we have the X -stable convergence

 L−s √  J,(l) n Ψk,T,n − 1 −→ J,(l)

on ΩT

J,(l)

Φk,T

(8.9)

kB (l) (g4 )T

. J,(l)

J,(l)

Here, the limit Φk,T /kBT

(g4 ) in (8.9) is by construction defined on the extended

). Further the statement of the X -stable convergence on  F, P probability space (Ω, J,(l)

the set ΩT

means that we have



√     J,(l)  g ΦJ,(l) /(kB (l) (g4 )T ) Y 1 J,(l) E g n(Ψk,T,n − 1) Y 1ΩJ,(l) → E k,T,n Ω T

T

8.1 Theoretical Results

215

for all bounded and continuous functions g and all X -measurable bounded random variables Y . For more background information on stable convergence in law see Appendix B. Example 8.9. Condition 8.7 is fulfilled in the setting of Poisson sampling discussed in Example 8.5. Part (i) is fulfilled by (5.2) and that part (ii) is fulfilled follows from Lemma 5.10 by choosing f (3) appropriately.  To construct a statistical test for deciding between ω ∈ ΩJ,(l) and ω ∈ ΩC,(l) under the null hypothesis that jumps do exist we need to assess the limit distribution in Theorem 8.8. For this purpose we use the methods discussed in Chapter 4. Let  n )n∈N and (Mn )n∈N be sequences of natural numbers which tend to infinity (K and define (l) ξˆk,n,m,− (s) = n

k−1 

(k − j)2 |I

j=1 (l) ξˆk,n,m,+ (s)

=n

k−1 

(l) (l)

(l)

in (s)+Vn,m (s)−j,1,n

|, (8.10)

2

(k − j) |I

j=1

(l) (l)

(l)

in (s)+Vn,m (s)+j,1,n

|

(l)

for m = 1, . . . , Mn where the random variable Vn,m (s) attains values in the set n, . . . , K  n } with probabilities {−K

P



(l) Vn,m (s)

 

˜S = |I (l) =k (l)

˜ in (s)+k,n

|





Kn 

|I

n ˜  =−K k

(l) (l) ˜  ,n in (s)+k

|

−1

,

(8.11)

ˆ(l) ˜ ∈ {−K n, . . . , K  n }. Here, (ξˆ(l) k k,n,m,− (s), ξk,n,m,− (s)) is chosen from the (l) (l) ˜   (ξk,n,− (tin (s)+k,n ˜ ), ξk,n,− (tin (s)+k,n ˜ )), k = −Kn , . . . , Kn , (l)

(l)

 n + 1 different realizations of (ξ which make up the 2K k,n,− (t), ξk,n,− (t)) which lie ”closest” to s, with probability proportional to the interval length |I

(l)

˜ in (s)+k,n (l)

|.

This corresponds to the probability with which a random variable which is uniformly (l) ˜ = −K  n , . . . , Kn,  but distributed on the union of the intervals I (l) , k ˜ in (s)+k,n

otherwise independent from the observation scheme would fall into the interval (l) I ˜ . Due to the structure of the predictable compensator ν the jump times in (s)+k,n (l)

Sp

of the Itˆ o semimartingale X (l) are also evenly distributed in time. This (l)

explains why we choose such a random variable Vn,m (s) for the estimation of (l) (l) (l) (l) (l) the law of (ξˆk,n,m,− (Sp ), ξˆk,n,m,− (Sp )). The random variables Vn,m (s) and

216

8 Testing for the Presence of Jumps

(l) (l) (l) (l) ξˆk,n,m,− (Sp ), ξˆk,n,m,− (Sp )) can be defined on the extended probability space

).  F, P (Ω, (l) Using the estimators (8.10) for realizations of ξk (s) we build the following J,(l)

estimators for realizations of Φk,T J,(l)

×

$





% Φ k,T,n,m = 4

(l)

Δi,n X (l)

3

(l)

(l)

1{|Δ(l) X (l) |>β|I (l) | } i,n

(l) i:ti,n ≤T

(l)

(l)

i,n

(l)

(l)

(l)

(l)

(l)

(˜ σn (ti,n , −))2 ξˆk,n,m,− (ti,n ) + (˜ σn (ti,n , +))2 ξˆk,n,m,+ (ti,n )Un,i,m , (8.12)

m = 1, . . . , Mn , where β > 0 and  ∈ (0, 1/2). Here an increment which is large compared to a given threshold, compare Section 2.3, is identified as a jump (l) (l) and the local volatility is approximated using the estimators σ ˜n (s, −), σ ˜n (s, +) (l) defined in Chapter 6. Further the Un,i,m are i.i.d. standard normal distributed random variables which are independent of F and can be defined on the extended ) as well.  F, P probability space (Ω, We then denote by

% % Q k,T,n (α) = Qα J,(l)



% Φ k,T,n,m |m = 1, . . . , Mn J,(l)





(8.13)



% the αMn -th largest element of the set Φ k,T,n,m |m = 1, . . . , Mn . J,(l)

% We will see that Q k,T,n (α) converges on ΩT J,(l)

J,(l)

J,(l) Qk,T (α)

under appropriate conditions

J,(l) Fk,T

to the X -conditional α-quantile of which is defined as the (under the upcoming condition unique) X -measurable [−∞, ∞]-valued random variable J,(l) Qk,T (α) fulfilling

    ΦJ,(l) ≤ QJ,(l) (α)X (ω) = α, ω ∈ ΩJ,(l) , P T k,T k,T 

J,(l)



J,(l) c

and for completeness we set Qk,T (α) (ω) = 0, ω ∈ (ΩT

J,(l)

) such that Qk,T (α)

J,(l)

is well defined on all of Ω. Such a random variable Qk,T (α) exists if Condition J,(l)

8.10 is fulfilled because the X -conditional distribution of Φk,T will be almost J,(l)

surely continuous on ΩT

under Condition 8.10.

Condition 8.10. Assume that Condition 8.7 is fulfilled and suppose that the (l) set {s ∈ [0, T ] : σs = 0} is almost surely a Lebesgue null set. Further, let the P

 n )n∈N and (Mn )n∈N are sequence (bn )n∈N fulfil |πn |T /bn −→ 0 and suppose that (K  n /n → 0. Additionally, sequences of natural numbers converging to infinity with K

8.1 Theoretical Results

217

(i) it holds

 (l) (l)   P((ξˆk,n,1,− (sp ), ξˆk,n,1,+ (sp )) ≤ xp , p = 1, . . . , P |S) P −

P 

  ((ξ (l) (sp ), ξ (l) (sp )) ≤ xp ) > ε → 0 P k,− k,+

p=1

as n → ∞, for all ε > 0 and any x = (x1 , . . . , xP ) ∈ R2×P , P ∈ N, and sp ∈ (0, T ), p = 1, . . . , P . (ii) The volatility process σ (l) is itself an Itˆ o semimartingale, i.e. a process of the form (1.1). C,(l)

(iii) On ΩT

J,(l)

J,(l)

we have kCk,T > CT

almost surely.

Part (i) of Condition 8.10 guarantees that the bootstrapped realizations (l) (l) (ξˆk,n,m,− (s), ξˆk,n,m,+ (s)) (l)

(l)

consistently estimate the distribution of (ξk,− (s), ξk,+ (s)) and thereby that the

% quantity Q k,T,n (α) yields a valid estimator for Qk,T (α) on ΩT J,(l)

J,(l)

J,(l)

. Part (ii) is

(l) (l) (l) (l) needed for the convergence of the volatility estimators σ ˜n (Sp , −), σ ˜n (Sp , +) (l) J,(l) for jump times Sp , and part (iii) guarantees that Ψk,T,n converges under the

alternative to a value different from 1, which is the limit under the null hypothesis. Theorem 8.11. If Condition 8.10 is fulfilled, the test defined in (8.6) with

% Q k,T,n (1 − α) J,(l)

J,(l) ck,T,n (α) = √

nkV (l) (g4 , πn )T

, α ∈ [0, 1],

has asymptotic level α in the sense that we have

    ΨJ,(l) > 1 + c J,(l) (α)F J,(l) → α, P k,T,n k,T,n

α ∈ [0, 1],

(8.14)

α ∈ (0, 1],

(8.15)

J,(l)

with P(F J,(l) ) > 0. for all F J,(l) ⊂ ΩT The test is consistent in the sense that we have

    ΨJ,(l) > 1 + c J,(l) (α)F C,(l) → 1, P k,T,n k,T,n C,(l)

for all F C,(l) ⊂ ΩT

with P(F C,(l) ) > 0.

218

8 Testing for the Presence of Jumps

Note that to carry out the test introduced in Theorem 8.11 the unobservable √ variable n is not explicitly needed, even though n occurs in the definition of J,(l) J,(l) ck,T,n (α). This factor actually cancels in the definition of ck,T,n (α) because it

%J,(l) (1 − α) as a linear factor. What remains is the dependence of also enters Q k,T,n

 n on n, though, but for these auxiliary variables only a rough idea of bn and K the magnitude of n usually is sufficient. Similar observations hold for all tests constructed later on in this chapter and in Chapter 9 as well. The simulation results in Section 8.2 show that the convergence in (8.14) is √ J,(l) rather slow, because certain terms in n(Ψk,T,n − 1) which vanish in the limit contribute significantly in the small sample. Our goal is to diminish this effect by including estimates for those terms in the testing procedure. The asymptotically vanishing terms stem from the continuous part which is mostly captured in the small increments. To estimate their contribution we define J,(l)

Ak,T,n = n



(l)

(Δi,k,n X)4 1{|Δ(l)

(l)

i,k,n X

i≥k:ti,n ≤T

− kn

 (l)

(l) ≤β|I (l)  i,k,n | }

(l)

(Δi,n X (l) )4 1{|Δ(l) X (l) |≤β|I (l) | } . i,n

i≥1:ti,n ≤T

i,n

using the same β,  as in (8.12). We then define for ρ ∈ (0, 1) the adjusted estimator n−1 Ak,T,n J,(l)

 J,(l) (ρ) = ΨJ,(l) − ρ Ψ k,T,n k,T,n

kV (l) (g4 , πn )T

where we partially correct for the contribution of the asymptotically vanishing terms. Corollary 8.12. Let ρ ∈ (0, 1). If Condition 8.10 is fulfilled, it holds with the notation from Theorem 8.11

 J,(l)   J,(l) J,(l)  Ψ   P → α, k,T,n (ρ) > 1 + ck,T,n (α) F J,(l)

for all F J,(l) ⊂ ΩT

C,(l)

(8.16)

α ∈ (0, 1],

(8.17)

with P(F J,(l) ) > 0 and

 C,(l)   J,(l) J,(l)  Ψ   P → 1, k,T,n (ρ) > 1 + ck,T,n (α) F

for all F C,(l) ⊂ ΩT

α ∈ [0, 1],

with P(F C,(l) ) > 0.

The closer ρ is to 1 the faster is the convergence in (8.16), but also the slower is the convergence in (8.17). Hence an optimal ρ should be chosen somewhere in between. Our simulation results in Section 8.2 show that it is possible to pick a ρ very close to 1 without significantly worsening the power compared to the test from Theorem 8.11.

8.1 Theoretical Results

219

Example 8.13. The assumptions on the observation scheme in Condition 8.10 are fulfilled in the Poisson setting. Part (iii) is fulfilled as shown in Example 8.5 and that part (i) is fulfilled follows from Lemma 5.12.  In fact for our testing procedure to work in the Poisson setting we do not need the weighting from (8.11). All intervals could also be picked with equal probability. This is due to the fact that the lengths (n|I

(l) (l)

(l)

in (s)+Vn,m (s)+j,n

|)j=−(k−1),...,−1,1,...,k−1

of intervals to the left and to the right of I (l) ξˆk,n,m,− (s),

(l) ξˆk,n,m,+ (s)

(l) (l)

(l)

in (s)+Vn,m (s),n

and hence the variables

are (asymptotically) independent of n|I

(l) (l)

(l)

in (s)+Vn,m (s),n

|.

However, the weighting is important if the interval lengths of consecutive intervals are dependent as illustrated in the following example. Example 8.14. Define an observation scheme by (l)

(l)

t2i,n = 2i/n,

t2i+1,n = (2i + 1 + α)/n,

i ∈ N0 , with α ∈ (0, 1); compare Example 3 in [8]. Let us consider the case k = 2. The observation scheme is illustrated in Figure 8.2. It can be easily checked that (l) [2],(l) Condition 8.2 holds with G4 (t) = (1 + α2 )t and G4 (t) = t. Further it can be shown similarly as in [8] that Condition 8.7 is fulfilled with ΓJ,(l) defined via ΓJ,(l) (s, {(1 + α, 1 + α)}) =

1−α , 2

ΓJ,(l) (s, {(1 − α, 1 − α)}) =

1+α 2

(l) for all s > 0. Hence in order for the distribution of ξˆk,n,1 (s) to approximate (l)

(l)

ΓJ,(l) (s, ·) the variable in (s) + Vn,m (s) has to pick the intervals of length (1 + α)/n with higher probability than those with length (1 − α)/n, because it holds n|I n|I

(l) (l)

(l)

(l) (l) | = 1 + α ⇒ ξˆ2,n,m,− (s) = ξˆ2,n,m,+ (s) = 1 − α,

(l)

| = 1 − α ⇒ ξˆ2,n,m,− (s) = ξˆ2,n,m,+ (s) = 1 + α.

in (s)+Vn,m (s),n (l) (l)

(l)

in (s)+Vn,m (s),n

(l)



220

8 Testing for the Presence of Jumps (1 − α)/n

(1 + α)/n 1/n

(l)

(l)

t1,n

5/n

(l)

t2,n = 2/n

(1 − α)/n

(1 + α)/n

3/n

(l)

t0,n = 0

(1 − α)/n

(1 + α)/n

(l)

t3,n

t4,n = 4/n

(l)

t5,n

(l)

t6,n = 6/n

Figure 8.2: The sampling scheme from Example 8.14

8.2 Simulation Results To verify the effectivity and to study the finite sample properties of the developed tests we apply them on simulated data. In our simulation the observation times originate from a Poisson process with intensity n which corresponds to λl = 1 in Example 8.5 and yields on average n observations in the time interval [0, 1]. We simulate from the model (l)

dXt

(l)

(l)

= Xt σdWt



(l)

+α R

Xt− xμ(l) (dt, dx)

(8.18)

(l)

where X0 = 1 and the Poisson measure μ(l) has the predictable compensator ν (l) (dt, dx) = κ

1[−h,−l]∪[l,h] (x) 2(h − l)

dtdx.

This model is a one-dimensional version of the model used in a similar simulation study in Section 6 of [32]. We consider the parameter settings displayed in Table 8.1 with σ 2 = 8 × 10−5 in all cases. We choose n = 100, n = 1,600 and n = 25,600. In a trading day of

Table 8.1: Parameter settings for the simulation.

Case I-j II-j III-j Cont

α 0.01 0.01 0.01 0.00

Parameters κ l h 1 0.05 0.7484 5 0.05 0.3187 25 0.05 0.1238

6.5 hours, this corresponds to observing X (l) on average every 4 minutes, every 15 seconds and every second. The cases I-j to III-j correspond to the presence of jumps of diminishing size. When there are smaller jumps we choose a situation

8.2 Simulation Results

221

where there are more jumps such that the overall contribution of the jumps to the quadratic variation is roughly the same in all three cases. The fraction of the quadratic variation which originates from the jumps matches the one estimated in real financial data from [46]. In all three cases where the model allows for jumps we only use paths in the simulation study where jumps were realized. In the fourth case Cont we consider a purely continuous model. The parameter values in Table 8.1 are taken from [32]. Regarding the free parameters in our testing procedures we set β = 0.03 and ω = 0.49. Here, we choose the same values as in [32] and thereby follow their recommendation to pick  close to 1/2 and β to be about 3 to 4 times the magnitude of σ (which in general is unknown but can easily be estimated from the data). (l) (l) This choice for β is reasonable as increments Δi,n X ≈ Δi,n C where the jump (l)

part is negligible are roughly normally distributed with variance σ 2 |Ii,n |. Thereby these increments are filtered out with high probability as a normal distributed √ random variable rarely exceeds 3 standard deviations. Further we use bn = 1/ n √  n = ln(n), Mn = 10 n in for the local interval in the estimation of σs and K √ (l) (l) the simulation of the ξˆk,n,m,− (s), ξˆk,n,m,+ (s). bn = 1/ n is chosen in the center of the allowed range between bn constant and bn = O(log(n)/n) which balances the benefits from choosing bn small (smaller bias in the estimation of σ if σ is not flat) and bn large (less variance in the estimation of σ). This choice is also close to the optimal one in the sense of Theorem 13.3.3 in [30] for the estimation of  n is chosen rather small to keep the computation time low, Mn is spot volatility. K chosen to be large enough to justify a reasonable approximation to the theoretical quantiles. In the simulation study the results were very robust to the choice of bn ,  n , Mn . K We applied the two testing procedures from Theorem 8.11 and Corollary 8.12 for k = 2, 3, 5 and the results are displayed in Figures 8.4 and 8.5. In Figure 8.4 the results from Theorem 8.11 are presented. In the left column the empirical rejection rates are plotted against the theoretical value of the test and in the right J,(l) column we show estimated density plots based on the simulated values of Φk,T,n . In Figure 8.5 we present the results from the test in Corollary 8.12 for ρ = 0.9 (for the choice of ρ = 0.9 see Figure 8.3) in the same way. The density plots here show  J,(l) (ρ). the estimated density of Φ k,T,n In Figure 8.4 we observe in the case Cont that the power of the test from Theorem 8.11 is very good for n = 1,600 and n = 25,600. Further the empirical rejection rates match the asymptotic values rather well for all considered values of k in the cases I-j and II-j at least for the highest observation frequency corresponding to n = 25,600. However in the case III-j there is severe over-rejection even for n = 25,600. In general we observe over-rejection in all cases. The empirical rejection rates match the asymptotic values better in the cases where there are on average larger jumps. Further the results are better the smaller k is. Note

222

8 Testing for the Presence of Jumps

that for n = 100 the cases III-j and Cont are not distinguishable using our test as the rejection curves for n = 100 in those two cases are almost identical. The J,(l) density plots show the convergence of Φk,T,n to 1 in the presence of jumps and to (k + 1)/2 under the alternative as predicted from Example 8.5. In Figure 8.5 we immediately see that the observed rejection rates from the test based on Corollary 8.12 match the asymptotic values much better than those from the test based on Theorem 8.11. Hence adjusting the estimator has a huge effect on the finite sample performance of the test. Here, the observed rejection rates match the asymptotic values quite well in the case III-j at least for n = 25,600 and in the cases I-j and II-j we get already for n = 1,600 very good results. The results in the case Cont show that the power remains to be very good. The  J,(l) (ρ) is more centered density plots show that under the presence of jumps Φ k,T,n

 around 1 than Φk,T,n . Under the alternative Φ k,T,n (ρ) clusters around the value 1+(1−ρ)(k −1)/2 which is much closer to 1 than (k +1)/2, but the observed values  J,(l) (ρ) still seem to be large enough such that they can be well distinguished of Φ k,T,n from 1 as can be seen from the high empirical rejection rate in the case Cont. Figure 8.3 illustrates how the performance of the test from Corollary 8.12 depends on the choice of the parameter ρ. For this purpose we investigate for k = 2 the empirical rejection rates in the cases III-j and Cont with n = 25,600 for the test with level α = 5% dependent on the choice of ρ. We plot for ρ ∈ [0, 1] the empirical rejection rate under the null hypothesis in the case III-j which serves as a proxy for the type-I error of the test together with one minus the empirical rejection rate under the alternative hypothesis in the case Cont which serves as a proxy for the type-II error of the test. Finally, we plot the sum of both error proxies to obtain an indicator for the overall performance of the test dependent on the choice of ρ. As expected we observe a decrease in the type-I error as ρ increases and an increase in the type-II error. While we observe an approximately linear decrease in the type-I error, the type-II error is equal to zero until 0.8, then slightly increases and starts to steeply increase at ρ = 0.9. In this example, the overall error is minimized for a relatively large value of ρ close to ρ = 0.9. Further, we carried out simulations for the same four parameter settings based on equidistant observation times ti,n = i/n. In this specific setting the test from Theorem 8.11 coincides with tests discussed in [2] and in Chapter 10.3 of [3]. The simulation results are presented in Figures 8.6, 8.7 and 8.8 in the same fashion as in Figures 8.4, 8.5 and 8.3 for the irregular observations. We observe that the results both from Theorem 8.11 and Corollary 8.12 based on irregular observations are not significantly worse than those obtained in the simpler setting of equidistant observation times. Especially we can conclude that the adjustment technique introduced for Corollary 8.12 cannot only be used to improve the finite sample performance of our test based on irregular observations but also can be used to improve existing tests in the literature which are based on equidistant observations. J,(l)

J,(l)

223

1.0

8.2 Simulation ulation Results

0.2

0.4

0.6

0.8

Level 1-Power (Level+(1-Power))

0.0

0.05

0.0

0.2

0.4

0.6

0.8

1.0

ρ

Figure 8.3: This graphic shows for k = 2, α = 5% and n = 25,600 the empirical rejection rate in the case Cont (dotted line) and 1 minus the empirical rejection rate in the case III-j (dashed line) from the Monte Carlo simulation based on Corollary 8.12 as a function of ρ ∈ [0, 1].

224

8 Testing for the Presence of Jumps II-j

I-j

II-j

0.5

1.0

0.0

0.5

1.0

6 0 0.0

0.5

1.0

Cont

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

III-j

1.5

2.0

2.5

3.0

2.0

2.5

3.0

Cont

0.0

0.5

1.0

0

0.0

0.0

2

2

4

4

0.5

0.5

6

6

8

8

1.0

III-j

10

0.0

0

0.0

0.0

2

2

4

4

6

0.5

0.5

8

10

8

12

10

14

1.0

1.0

I-j

0.0

0.5

1.0

(a) Rejection curves for k = 2.

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

1.5

(b) Density estimation of

II-j

I-j

J,(l) Φ2,T,n .

II-j

0.5

1.0

0.0

0.5

1.0

0 0.0

0.5

1.0

Cont

1.5

2.0

2.5

3.0

0.0

0.5

1.0

III-j

1.5

2.0

2.5

3.0

Cont

0.0

0.5

1.0

3 0.0

0.5

1.0

(c) Rejection curves for k = 3.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

1

2

(d) Density estimation of I-j

II-j

3

4

J,(l) Φ3,T,n .

II-j

0.5

0.0

1.0

0.5

1.0

1.5

2.0

2.5

3.0

0.5

1.0

1.5

2.0

2.5

3.0

4

5

6

Cont

1.5

3 0.0

0.0

0.5

1

1.0

2

0.5

0.5

0.0

III-j

Cont

1.0

1.0

0.0

2.5

1.0

2.0

0.5

III-j

4

0.0

0

0

0.0

0.0

1

1

2

2

0.5

0.5

3

3

4

4

5

1.0

1.0

I-j

0

0

0.0

0.0

1

1

2

2

3

0.5

0.5

4

4

5

6

1.0

1.0

III-j

5

0.0

0

0.0

0.0

1

2

2

3

4

0.5

0.5

4

6

5

6

8

1.0

1.0

I-j

0.0

0.0

0.5

1.0

0.0

0.5

(e) Rejection curves for k = 5.

1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

1

2

3

(f ) Density estimation of

J,(l) Φ5,T,n .

Figure 8.4: These graphics show the simulation results for the test from Theorem 8.11. The dotted lines correspond to n = 100, the dashed lines to n = 1,600 and the solid lines to n = 25,600. In all cases N = 10,000 paths were simulated.

8.2 Simulation Results II-j

I-j

II-j

6

8

0.5

1.0

0.0

0.5

1.0

0 0.0

0.5

1.0

Cont

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

III-j

1.5

2.0

2.5

3.0

2.0

2.5

3.0

Cont

30

6

0.0

0.5

1.0

0.0

0.5

1.0

(a) Rejection curves for k = 2.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

1.5

 J,(l) (0.9). (b) Density estimation of Φ 2,T,n

II-j

I-j

II-j

1.0

4 0.5

1.0

0 0.0

0.5

1.0

Cont

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

(c) Rejection curves for k = 3.

1.5

2.0

2.5

3.0

30 25 15 10 5 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

1

2

(d) Density estimation of

II-j

I-j

3

4

 J,(l) (0.9). Φ 3,T,n

II-j

0.5

1.0

0.0

0.5

1.0

0 0.0

0.5

1.0

Cont

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

III-j

1.5

2.0

2.5

3.0

4

5

6

Cont

0.0

0.5

1.0

0.0

0.5

(e) Rejection curves for k = 5.

1.0

0

0

0.0

0.0

1

5

2

10

0.5

0.5

3

15

4

1.0

III-j

20

0.0

0

0.0

0.0

1

1

2

2

3

0.5

0.5

3

4

4

5

1.0

1.0

I-j

1.0

20

5 4 2 1 1.0

0

0.0 0.5

0.5

Cont

3

0.5

0.5 0.0 0.0

0.0

III-j

6

1.0

0.0

35

0.5

III-j

7

0.0

0

0.0

0.0

1

2

2

3

4

0.5

0.5

6

5

6

8

7

1.0

1.0

I-j

0

0

0.0

0.0

2

10

20

4

0.5

0.5

40

8

50

10

1.0

III-j

60

0.0

0

0.0

0.0

2

2

4

4

6

0.5

0.5

10

8

12

10

14

1.0

1.0

I-j

225

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

1

2

3

(f ) Density estimation of

 J,(l) (0.9). Φ 5,T,n

Figure 8.5: These graphics show the simulation results for the test from Corollary 8.12. The dotted lines correspond to n = 100, the dashed lines to n = 1,600 and the solid lines to n = 25,600. In all cases N = 10,000 paths were simulated.

226

8 Testing for the Presence of Jumps II-j

I-j

II-j

1.0

0.5

1.0

0 0.0

0.5

1.0

Cont

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

III-j

1.5

2.0

2.5

3.0

2.0

2.5

3.0

Cont

0.0

0.5

1.0

0

0.0

0.0

2

2

4

4

6

0.5

0.5

6

8

8

1.0

0.0

12

0.5

III-j

10

0.0

0

0.0

0.0

2

2

4

4

6

0.5

0.5

6

8

8

10

1.0

1.0

I-j

0.0

0.5

1.0

(a) Rejection curves for k = 2.

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

1.5

(b) Density estimation of

II-j

I-j

J,(l) Φ2,T,n .

II-j

0.0

0.5

1.0

0.0

0.5

1.0

0 0.0

0.5

1.0

Cont

1.5

2.0

2.5

3.0

0.0

0.5

1.0

III-j

1.5

2.0

2.5

3.0

Cont

0.5

1.0

3 0.0

0.5

1.0

(c) Rejection curves for k = 3.

0.5

1.0

1.5

2.0

2.5

3.0

0

1

2

(d) Density estimation of I-j

II-j

3

4

J,(l) Φ3,T,n .

II-j

5

1.0

0.0

0.5

1.0

0.0

0.5

0.0

1.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

III-j

Cont

1.5

2.0

2.5

3.0

4

5

6

Cont

2

0.0

0.5

1.0

0.5 0.0

0

0.0

0.0

1

1.0

0.5

0.5

1.5

3

2.0

1.0

1.0

III-j

0

0

0.0

0.0

1

1

2

2

0.5

0.5

3

3

4

1.0

I-j

0.0

4

0.0

0

0.0

0.0

1

1

2

2

3

0.5

0.5

4

4

5

5

6

1.0

1.0

III-j

0

0.0

0.0

1

2

2

3

4

0.5

0.5

4

5

6

6

1.0

1.0

I-j

0.0

0.0

0.5

(e) Rejection curves for k = 5.

1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

1

2

3

(f ) Density estimation of

J,(l) Φ5,T,n .

Figure 8.6: Simulation results for the test from Theorem 8.11 based on (l) equidistant observations ti,n = i/n. The dotted lines correspond to n = 100, the dashed lines to n = 1,600 and the solid lines to n = 25,600. In all cases N = 10,000 paths were simulated.

8.2 Simulation Results II-j

I-j

II-j

1.0

0.5

1.0

8 2 0 0.0

0.5

1.0

Cont

1.0

1.5

2.0

2.5

3.0

0.5

1.0

1.5

2.0

2.5

3.0

2.0

2.5

3.0

Cont

100

60

0.5

1.0

0.0

0.5

1.0

(a) Rejection curves for k = 2.

0 0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

1.5

 J,(l) (0.99). (b) Density estimation of Φ 2,T,n

II-j

I-j

II-j

1.0

5 4

0.5

1.0

0 0.0

0.5

1.0

Cont

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0

III-j

1.5

2.0

2.5

3.0

Cont

30

0.0

0.5

1.0

0.0

0.5

1.0

(c) Rejection curves for k = 3.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

1

2

(d) Density estimation of

II-j

I-j

3

4

 J,(l) (0.99). Φ 3,T,n

II-j

0.0

0.5

1.0

0.0

0.5

1.0

3 0 0.0

0.5

1.0

Cont

1.5

2.0

2.5

3.0

0.0

0.5

1.0

III-j

1.5

2.0

2.5

3.0

4

5

6

Cont

0.0

0.5

1.0

15 0.0

0.5

(e) Rejection curves for k = 5.

1.0

5 0

0

0.0

0.0

5

10

10

0.5

0.5

15

20

20

25

1.0

1.0

III-j

0

0.0

0.0

1

1

2

2

3

0.5

0.5

4

5

4

1.0

1.0

I-j

0

0

0.0

0.0

10

20

20

40

0.5

0.5

60

40

1.0

0.0

80

0.5

III-j

50

0.0

0

0.0

0.0

1

2

2

3

4

0.5

0.5

6

6

8

1.0

1.0

I-j

0.0

7

0.0

0

0.0

0.0

20

50

40

0.5

0.5

0.0

III-j

80

1.0

0.0

150

0.5

III-j

100

0.0

0

0.0

0.0

2

4

4

6

6

0.5

0.5

8

10

10

12

12

1.0

1.0

I-j

227

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0

1

2

3

(f ) Density estimation of

 J,(l) (0.99). Φ 5,T,n

Figure 8.7: Simulation results for the test from Corollary 8.12 based on (l) equidistant observations ti,n = i/n. The dotted lines correspond to n = 100, the dashed lines to n = 1,600 and the solid lines to n = 25,600. In all cases N = 10,000 paths were simulated.

8 Testing T for the Presence o of Jumps

1.0

228

0.2

0.4

0.6

0.8

Level 1-Power (Level+(1-Power))

0.0

0.05

0.0

0.2

0.4

0.6

0.8

1.0

ρ

Figure 8.8: This graphic shows for k = 2, α = 5%, n = 25,600 and equidistant (l) observations ti,n = i/n the empirical rejection rate in the case Cont (dotted line) and 1 minus the empirical rejection rate in the case III-j (dashed line) from the Monte Carlo simulation based on Corollary 8.12 as a function of ρ ∈ [0, 1]. We achieve a minimal overall error for approximately ρ = 0.99.

8.3 The Proofs J,(l)

Proof of Theorem 8.3. The convergence in (8.4) on ΩT follows from Corollary 2.2 and Theorem 2.40 as discussed before Condition 8.2. Further the convergence C,(l) J,(l) on ΩT follows from (8.3) since CT > 0 almost surely holds because of by

T 0

(l)

(l)

|σs |ds > 0 almost surely and because G4 is strictly increasing by Condition (l)

(l)

(l)

8.2. To prove (8.3) note that it holds Xt 1ΩC,(l) = (Bt + Ct )1ΩC,(l) with T

T

8.3 The Proofs

229



(l)

Bt since

T 0

t

= 0



(l)

bs −



R2

δ (l) (s, z)1{δ(s,z)≤1} λ(dz) ds

δ (l) (s, z)μ(ds, dz) ≡ 0 on ΩT

C,(l)

(8.19)

. (8.3) then follows by applying Corol(l)

(l)

lary 2.19 and Theorem 2.41(a) to the process Bt + Ct . Proof of Theorem 8.8. We prove  L−s J,(l) √  (l) n V (g4 , [k], πn )T − kV (l) (g4 , πn )T −→ Φk,T .

(8.20) P

The central limit theorem (8.9) then follows easily from (8.20), V (l) (g4 , πn )T −→ B (l) (g4 )T , Proposition B.7(i) and Lemma B.5. (l) (r, q) of σ (l) and C (l) Step 1. Using the discretized versions σ ˜ (l) (r, q) and C introduced in Step 1 in the proof of Theorem 3.2 we define 

√ R(l) (n, q, r) = 4 n

(l)

(Δi,n N (l) (q))3

(l)

k−1 

(l)

(l)

(l) (r, q) + Δ (l) (k − j)(Δi+j,n C i−j,n C (r, q)).

j=1

i:ti,n ≤T

and show

√   lim lim sup lim sup P  n V (l) (g4 , [k], πn )T − kV (l) (g4 , πn )T

q→∞ r→∞

n→∞





− R(l) (n, q, r) > ε = 0

(8.21)

for all ε > 0. As the proof of (8.21) is rather lengthy and technical we will first discuss the complete rough structure of the proof of Theorem 8.8 and present the proof of (8.21) afterwards. Step 2. Next we prove L−s



R(l) (n, q, r) −→ Φk,T (q, r) := 4 J,(l)

(ΔN (l) (q)S (l) )3 q,p

(l)

Sq,p ≤T



(l)

(l)

(l)

(l)

× (˜ σ (l) (r, q)S (l) − )2 ξk,− (Sq,p ) + (˜ σ (l) (r, q)S (l) )2 ξk,+ (Sq,p ) q,p

1/2

U

q,p

(l) (l)

Sq,p

(8.22)

as n → ∞ for all q > 0 and r ∈ N. To this end note that on the set Ω(l) (n, q, r) where two different jumps of N (l) (q) are further apart than |πn |T and any jump time of N (l) (q) is further away from j2−r than k|πn |T for any j ∈ {1, . . . , T 2r } it holds



R(l) (n, q, r)1Ω(l) (n,q,r) = 4

√  (ΔN (l) (q)S (l) )3 n (k − j) k−1

q,p

(l)

j=1

Sq,p ≤T

×(˜ σ (l) (r, q)S (l) − Δ q,p

(l) (l)

(l)

in (Sq,p )−j,n

W

(l)

+˜ σ (l) (r, q)S (l) Δ q,p

(l) (l)

(l)

in (Sq,p )+j,n

W

(l)

)1Ω(l) (n,q,r) ,

230

8 Testing for the Presence of Jumps

compare (3.23) for the definition of W . Comparing the proof of (3.25) we find that Condition 8.7(ii) yields the X -stable convergence

√ k−1  n

(k −j)Δ

j=1

(l) (l) (l) in (Sq,p )−j,n

L−s

−→

 $

W

(l)

,

k−1 √  (l) n (k −j)Δ (l)

(l) in (Sq,p )+j,n

j=1

(l) (l) (l) ξk,− (Sq,p )U (l) , S ,−

$

(l)

(l)

W

(l)

ξk,+ (Sq,p )USq,p ,+

q,p

(l)



(l)  (l) Sq,p ≤T



 (l) Sq,p ≤T

(l)

for standard normally distributed random variables (Us,− , Us,+ ) which are inde(l)

(l)

pendent of F and of the ξk,− (s), ξk,+ (s). Using this stable convergence, Lemma B.3 and the continuous mapping theorem for stable convergence stated in Lemma B.5 we then obtain 4



√  (ΔN (l) (q)S (l) )3 n (k − j) k−1

q,p

(l)

j=1

Sq,p ≤T

× (˜ σ (l) (r, q)S (l) − Δ q,p

L−s

(l) (l)

(l)

in (Sq,p )−j,n

W

(l)

(l)

+σ ˜ (l) (r, q)Sq,p Δ

(l) (l)

(l)

in (Sq,p )+j,n

W

(l)

)

J,(l)

−→ Φk,T (q, r).

Because of P(Ω(l) (n, q, r)) → 1 as n → ∞ for any q, r > 0 this convergence yields (8.22). Step 3. Finally we need to show

(|Φ lim lim sup P k,T − Φk,T (q, r)| > ε) = 0, J,(l)

J,(l)

(8.23)

q→∞ r→∞

for all ε > 0. (8.23) can be proven using the fact that ΓJ,(l) (·, dy) has uniformly bounded first moments together with the boundedness of the jump sizes of X (l) respectively N (l) (q). Step 4. Finally by combining (8.21)–(8.23) and using Lemma B.6 we obtain (8.20). (l)

(l)

Proof of (8.21). To simplify notation we set Δi,k ,n X = 0, Ii,k ,n = ∅, k = 1, k, (l)

whenever ti,n > T in this proof. Step 1. We first prove  √    lim sup P  n V (l) (g4 , [k], πn )T − kV (l) (g4 , πn )T − R(l) (n) > ε = 0 (8.24) n→∞

where R(l) (n) =

√

n

 (l)

i:ti,n ≤T

(l)

4(Δi,n X (l) )3

k−1  j=1



(l)

(l)



(k − j) Δi−j,n X (l) + Δi+j,n X (l) .

8.3 The Proofs

231

Using the identity (x1 + . . . + xk )4 =

k 

(xi )4 + 4

(xi )3 xj + 6

i=1 j =i

i=1

+ 12

k  

k  



k  

(xi )2 (xj )2

i=1 j>i

(xi )2 xj xj  + 24

i=1 j =i j  >j

k    

x i x i x j x j 

i=1 i >i j>i j  >j

(8.25) which is a specific case of the multinomial theorem, we derive √

n(V (l) (g4 , [k], πn )T − kV (l) (g4 , πn )T )

=

√



n

k−1 



(l)

(l)

(l)

(l)

(k − j) 4(Δi,n X (l) )3 Δi−j,n X (l) + 4(Δi,n X (l) )3 Δi+j,n X (l)



(l) i:ti,n ≤T j=1

+ OP

√

n



k−1 

(l) i:ti,n ≤T

j=1

(l)



(l)

K(Δi,n X (l) )2 (Δi+j,n X (l) )2 + OP

√

n|πn |T



(8.26)

where we used the inequalities |xi x i | ≤ (xi )2 + (xi )2 , |xj xj  | ≤ (xj )2 + (xj  )2 to include terms with powers (2, 1, 1, 0) and (1, 1, 1, 1) from (8.25) into the first summand of the last line of (8.26). The last term in (8.26) is due to boundary effects at T . By iterated expectations and inequality (1.12) we get √

nE





k−1 

(l) i:ti,n ≤T

j=1

(l)

  

(l)

K(Δi,n X (l) )2 (Δi+j,n X (l) )2 S ≤

√

nKT |πn |T .

√ Hence this quantity and the OP ( n|πn |T )-term vanish for n → ∞ by Condition 8.7(i). This observation yields (8.24). Step 2. Next we prove lim lim sup P(|R(l) (n) − R(l) (n, q)| > ε) → 0

(8.27)

q→∞ n→∞

where R(l) (n, q) =

√

n

 (l)

i:ti,n ≤T

(l)

4(Δi,n N (l) (q))3

k−1  j=1



(l)

(l)



(k − j) Δi−j,n C (l) + Δi+j,n C (l) .

232

8 Testing for the Presence of Jumps

Therefore we first consider

(l) (n, q))| > ε) → 0 lim lim sup P(|R(l) (n) − R

(8.28)

q→∞ n→∞

for all ε > 0 with

(l) (n, q) = R

√



n

4(Δi,n N (l) (q))3

(l)

k−1 



(l)

(l)

(k − j) Δi−j,n X (l) + Δi+j,n X (l)



j=1

i:ti,n ≤T

Using |a3 − b3 | = |a − b||a2 + ab + b2 | ≤

3 |a − b|(a2 + b2 ) 2

we derive

(l) (n, q)| |R(l) (n) − R   (l) (l)   √ Δ X − Δ(l) N (l) (q) (Δ(l) X (l) )2 + (Δ(l) N (l) (q))2 ≤ nK i,n i,n i,n i,n (l)

i:ti,n ≤T

×

k−1 

(l)

(l)

|Δi−j,n X (l) + Δi+j,n X (l) |.

(8.29)

j=1

The S-conditional expectation of (8.29) is using iterated expectations, the H¨ older in(l) (l) (l) equality with p1 = 3, p2 = 3/2 on Δi,n (B (l) (q)+C (l) )((Δi,n X (l) )2 +(Δi,n N (l) (q))2 ), (l)

(l)

(l)

the Cauchy-Schwarz inequality on Δi,n M (l) (q)((Δi,n X (l) )2 + (Δi,n N (l) (q))2 ) and Lemma 1.4 bounded by √

nK



(l)

(l)

(l)

(l)

(l)

(l)

(l)

(l)

+ (eq |Ii,n |)1/2 (|Ii,n | + K|Ii,n | + Kq |Ii,n |4 )1/2 k−1 

×

(l)

(Kq |Ii,n |3 + |Ii,n |3/2 )1/3 (|Ii,n | + K|Ii,n | + Kq |Ii,n |3 )2/3

(l) i:ti,n ≤T



(l)

|Ii+j,n |1/2

j=−(k−1),j =0

≤ K(Kq (|πn |T )1/6 + (eq + Kq (|πn |T )3 )1/2 ) ×

√

n

 (l)

i:ti,n ≤T

(l)

|Ii,n |

k−1 

(l)

|Ii+j,n |1/2

j=−(k−1),j =0

√ ≤ K(Kq (|πn |T )1/6 + (eq )1/2 ) n

 (l) i:ti,n ≤T

(l)

(l)

|Ii,k,n ||Ii,k,n |1/2

8.3 The Proofs

233





≤ K(Kq (|πn |T )1/6 + (eq )1/2 ) nT

(l)

|Ii,k,n |2

1/2 (8.30)

(l)

i:ti,n ≤T (l)

(l)

(l)

where we used |Ii,n ||Ii+j,n |1/2 ≤ |Imax{i,i+j},k,n |3/2 for |j| ≤ k − 1 and the Cauchy-Schwarz inequality for sums to obtain the last two inequalities. The last [k],(l) bound vanishes as first n → ∞ and then q → ∞ because of G4 (T ) = OP (1) by Condition 8.2. Hence by Lemma 2.15 we have proved (8.28). Further we prove

(l) (n, q))| > ε) → 0 lim lim sup P(|R(l) (n, q) − R

(8.31)

q→∞ n→∞

for all ε > 0. Denote by Ω(l) (n, q) the set where two jumps of N (l) (q) are further apart than k|πn |T . By iterated expectations and Lemma 1.4 it holds

(l) (n, q))|1Ω(l) (n,q) |S] E[|R(l) (n, q) − R √ ≤K n





 

(l)

|Δi+j,n (B (l) (q) + M (l) (q))|S

j=−(k−1),j =0

(l)

i:ti,n ≤T

√ ≤K n

k−1 

(l) E |Δi,n N (l) (q)|3

(l)

k−1 

(l)

(|Ii,n | + Kq |Ii,n |3 )

(l)

(l)

(Kq |Ii,n |2 + eq |Ii,n |)1/2

j=−(k−1),j =0

(l) i:ti,n ≤T





≤ K(1 + Kq (|πn |T )2 )(Kq |πn |T + eq )1/2 nT

(l)

|Ii,k,n |2

1/2

(l)

i:ti,n ≤T

where the last inequality follows as in (8.30). The last bound vanishes as first n → ∞ and then q → ∞ by Condition 8.2. Hence by Lemma 2.15 we have proved (8.31) because of P(Ω(l) (n, q)) → 1 as n → ∞ for any q > 0 which together with (8.28) yields (8.27). Step 3. Finally we consider lim lim sup lim sup P(|R(l) (n, q) − R(l) (n, q, r)| > ε) → 0.

q→∞ r→∞

(8.32)

n→∞

Using iterated expectations, inequality (1.11) and inequality (2.1.34) from [30] we obtain E[|R(l) (n, q) − R(l) (n, q, r)||S]

≤ (K + Kq (|πn |T )2 ) ×E

√

n

 (l)

i:ti,n ≤T

(l)

|Ii,n |

k−1 



(

j=−(k−1),j=0

(l)

ti+j,n (l) ti+j−1,n

  |σs(l) − σ ˜ (l) (r, q)s |2 ds)1/2 S

234

8 Testing for the Presence of Jumps ≤ (K + Kq (|πn |T )2 )E

√



n

(l)

|Ii,n |(2k − 2)1/2

(l) i:ti,n ≤T

×



k−1 



(l)

ti+j,n (l) ti+j−1,n

j=−(k−1),j=0

|σs(l) − σ ˜ (l) (r, q)s |2 ds

1/2   S .

This quantity is using the Cauchy-Schwarz inequality for sums further bounded by (K + Kq (|πn |T )2 )

$

(l),n

G4

(T )





× E (2k − 2)2



(l)

i:ti,n ≤T 2

−→ K

$

 (l) G4 (T )E

(l)

 (l),n G4 (T )E



T 0

(l) |σs

(l)

|σs − σ ˜ (l) (r, q)s |2 ds

1/2   S

ti−1,n

$

= (K + Kq (|πn |T ) ) P

(l)

ti,n



T 0

(l)

|σs − σ ˜ (l) (r, q)s |2 ds

−σ ˜ (l) (r, q)s |2 ds

1/2   S

1/2   S

where the convergence as n → ∞ follows from Condition 8.2. Here the limit vanishes as r → ∞ for any q > 0 since the expectation vanishes as r → ∞ by dominated convergence because σ (l) , σ ˜ (l) (r, q) are bounded by assumption and because (˜ σ (l) (r, q))r∈N is a sequence of right continuous elementary processes approximating σ (l) . Hence (8.32) follows from Lemma 2.15. Step 4. Combining (8.24), (8.27) and (8.32) we obtain (8.21). The structure of the upcoming proof of (8.14) in Theorem 8.11 and especially of (8.33) therein is identical to the structure of the proof of Theorem 4.3 and (4.5) in Chapter 4. Proof of Theorem 8.11. We first discuss the main ideas for the proof while more technical details will be proved afterwards. For proving (8.14) we need to show

    √n(V (l) (g4 , [k], πn )T − kV (l) (g4 , πn )T ) > Q %J,(l) (1 − α)F J,(l) → α P k,T,n which follows from Theorem 8.8 and



J,(l) 

 {|Q % lim P k,T,n (α) − Qk,T (α)| > ε} ∩ ΩT

n→∞

J,(l)

J,(l)

→0

(8.33)

for all ε > 0 and any α ∈ [0, 1]. For more details compare the proof of Theorem 4.3. Hence it remains to prove (8.33) for which we will give a proof after we discussed the main idea for the proof of (8.15).

8.3 The Proofs

235 J,(l)

C,(l)

For proving (8.15) we observe that Φk,T,n converges on ΩT to a limit strictly greater than 1 by Theorem 8.3 and Condition 8.10(ii). Hence it is sufficient to prove J,(l) ck,T,n (α)1ΩC,(l) = oP (1), α > 0,

(8.34)

T

to obtain (8.15). The proof of (8.34) will be given after the proof of (8.33). Similarly as in Chapter 4 we first prove the following proposition needed in the proof of (8.33). Proposition 8.15. Suppose Condition 8.10 is fulfilled. Then it holds

 P

Mn   1   1 {Φ  J,(l)

Mn

m=1





J,(l) 

(Φ  −P k,T ≤ Υ|X ) > ε ∩ ΩT J,(l)

k,T ,n,m ≤Υ}

→0

(8.35)

for any X -measurable random variable Υ and all ε > 0. (l)

Proof. Denote by Sq,p , p ∈ N, an increasing sequence of stopping times which exhausts the jump times of N (l) (q) and introduce the notation (l)

Yk (P, n, m) =



P 

(Δi(l) (S (l) ),n X)3 1{|Δ (l)

p=1

n

X|>β|I (l) (l) | (l) in (Sq,p ),n in (Sq,p ),n

q,p

(l)

(l)

× (˜ σn (ti(l) (S (l) ),n , −))2 ξˆk,n,m,− (t n

(l)

q,p

+ (˜ σn (ti (l)

Yk (P ) =

P 

(l) n (Sq,p ),n

(l)

(l)

(l) (l)

, +))2 ξˆk,n,m,+ (t



(ΔXSp )3 (σ

p=1

(l) (l)

Sq,p −

(l)

(l)

in (Sq,p ),n

)

(l) (l)

(l)

in (Sq,p ),n (l)

}

)2 ξk,− (Sq,p ) + (σ

)

1/2

(l) (l)

Sq,p

U

(l) 1 (l) , (l) n,in (Sq,p ),m {Sq,p ≤T } (l)

(l)

)2 ξk,+ (Sq,p )

1/2

U

(l) (l)

Sq,p

× 1{S (l) ≤T } . q,p

p %n,m Step 1. By specifying An,p , Ap , Z (s), Z p (s) and the function ϕ in a similar fashion like in Step 1 in the proof of Proposition 4.10 we obtain from Lemma 4.9, Condition 8.7 and Corollary 6.4 the convergence

 P

Mn    1 

 (Y (l) (P ) ≤ Υ|X ) > ε ∩ ΩJ,(l) → 0 (8.36)  1{Y (l) (P,n,m)≤Υ} − P T k

Mn

m=1

k

as n → ∞ for any fixed P .

236

8 Testing for the Presence of Jumps

Step 2. Next we show

 lim lim sup P

P →∞ n→∞

Mn   1   (1{Y (l) (P,n,m)≤Υ} − 1{Φ  J,(l)

Mn

k

m=1



k,T ,n,m ≤Υ}



J,(l) 

)  > ε ∩ ΩT

=0

(8.37) for all ε > 0 which follows from lim lim sup

P →∞ n→∞

Mn 1  (l) % J,(l) | > ε) = 0 P(|Yk (P, n, m) − Φ k,T,n,m Mn

(8.38)

m=1

for all ε > 0; compare Step 4 in the proof of Proposition 4.10. On the set Ω(l) (q, P, n) on which there are at most P jumps of N (l) (q) in [0, T ] and two different jumps of N (l) (q) are further apart than |πn |T it holds (l)

% |Yk (P, n, m) − Φ k,T,n,m |1Ω(l) (q,P,n) J,(l)



≤

(l) (l) |Δi,n (X (l) − N (l) (q))|3 1{|Δ(l) X (l) |>β|I (l) | } (ξˆk,n,m (i))1/2

(l)

i:ti,n ≤T

×

i,n

2 bn



≤K

 (l)

i,n



(l)

2 1/2

(l)

|Un,i,m |

(l)

j =i:Ij,n ⊂[ti,n −bn ,ti,n +bn ]

     (l) (l) Δ (B (q) + C (l) )3 + Δ(l) M (l) (q)3 (ξˆ(l) i,n

1 bn

k,n,m (i))

i,n

(l) i:ti,n ≤T

×

(l)

Δj,n X (l)

 (l)



(l)

(l)

Δj,n X (l)

2 1/2

(l)

|Un,i,m |

1/2

(8.39)

(l)

j =i:Ij,n ⊂[ti,n −bn ,ti,n +bn ]

with (l) (l) (l) (l) (l) ξˆk,n,m (i) = ξˆk,n,m,− (ti,n ) + ξˆk,n,m,+ (ti,n ).

For the continuous parts in (8.39) we get using the Cauchy-Schwarz inequality, (l) (l) Lemma 1.4 and ξˆk,n,m,− (ti,n ), ξˆk,n,m,+ (ti,n ) ≤ nK|πn |T

E





(l)

|Δi,n (B (l) (q) + C (l) )|3

1

(l)

i:ti,n ≤T

≤K

 (l)

i:ti,n ≤T



bn



(l)

(l)

(l)

(l)

Kq |Ii,n |6 + |Ii,n |3

 2bn bn

 1/2 √ = K n|πn |T Kq (|πn |T )3 + 1 T

(l)

j =i:Ij,n ⊂[ti,n −bn ,ti,n +bn ]

(l) (l) (l) (l) × ξˆk,n,m,− (ti,n ) + ξˆk,n,m,+ (ti,n )



(l)

(Δj,n X (l) )2

n|πn |T

1/2

1/2

(l)

  

|Un,i,m |S

1/2

8.3 The Proofs

237

which vanishes as n → ∞ by Condition 8.7(i). (l) Furthermore we get for the remaining terms containing |Δi,n M (l) (q)|3 in (8.39) using inequalities (1.10) and (1.12) E

 (l)

i:ti,n ≤T

 (l) (l) 3  1 Δ M (q) i,n bn

(l)

k,n,m (i)

1/2

(l)

   (l) |Un,i,m |S

1 bn

 1 (l) + E |Δi,n M (l) (q)|3 E bn

 1/2   (l) × E ξˆk,n,m (i)S  (l)

i:ti,n ≤T

(l)

(l)

j:Ij,n ⊂[ti,n −bn ,ti,n )

(l)



(l)

1/2    S

(l)

Δj,n X (l)

(l)

  2 1/2   S, Ft(l) S i,n

 1/2  (l) (Δj,n X (l) )2 S

 (l)

(l)

(l)

j:Ij,n ⊂[ti,n −bn ,ti,n )

 1/2     (l) (Δj,n X (l) )2 S, Ft(l) S

 (l)

(l)

(l)

i,n

j:Ij,n ⊂[ti,n ,ti,n +bn ]

n K  b 1/2    k−1 1/2  n (l) (l) |Ii,n | K n (k − j)2 (|Ii+k−j,n | + |Ii+k+j,n |) ˜ ˜ bn j=1 ˜  k=−Kn

(l)

× |Ii+k,n ˜ | 

n K 



˜  =−K n k

(l)

|Ii,n |



(l)

|Ii,n |

(l)

i:ti,n ≤T

+ OP ( n|πn |T )

(l)

|Ii+k˜ ,n |

n K  ˜ n k=− K

(l)

i:ti,n ≤T

√

(l)

(Δj,n X (l) )2

j:Ij,n ⊂[ti,n ,ti,n +bn ]

i,n

 1 (l) + E |Δi,n M (l) (q)|3 E bn





 (l) 1/2   S × E ξˆk,n,m (i)  

1  (l) ≤ Keq |Ii,n | E bn i:t ≤T

√ = Keq n

]

i−1,n

×

√ ≤ Keq n

1/2

(l)

 (l) E E[|Δi,n M (l) (q)|3 |S, Ft(l)

i:ti,n ≤T

≤ Keq

(l)

(l)

(Δj,n X (l) )2

j=i:Ij,n ⊂[ti,n −bn ,ti,n +bn ]

 (l) × ξˆ



≤K



(l)

|Ii+k,n ˜ |

k−1  j=−(k−1)

(l)

−1 



n K 

|Ii+k˜ ,n |

−1

˜  =−K n k

|Ii+j,n |1/2

n K  ˜ n k=− K

k−1 

(l)

|Ii+k+j,n |1/2 ˜

j=−(k−1)

(l)

|Ii+k,n ˜ |



n K  ˜  =−K n k

(l)

|Ii+k+ ˜ k ˜  ,n |

−1

(8.40)

238

8 Testing for the Presence of Jumps

˜ to derive the last equality. Then (8.40) where we changed the index i → i + k vanishes as in (8.30) because of 

Kn 

|Ii+k,n ˜ |



Kn 



n ˜ K k=−

|Ii+k+ ˜ k ˜  ,n |

−1

≤2

n ˜  =−K k

which we obtain as in (4.23). Hence by Lemma 2.15 we get (8.38) because of P(Ω(l) (q, P, n)) → ∞ as P, n → ∞ for any q > 0. Step 3. Finally we show P  J,(l) (Y (l) (P ) ≤ Υ|X )1 J,(l) −→ P P(Φk,T ≤ Υ|X )1ΩJ,(l) k Ω T

P

(l)

(8.41)

T

J,(l)

as P → ∞. We have Yk (P ) −→ Φk,T as P → ∞ and it can be shown that J,(l)

this convergence yields (8.41) since the X -conditional distribution of Φk,T is by Condition 8.7 almost surely continuous; compare Step 3 in the proof of Proposition 4.10 where a similar claim is proven in more detail. Step 4. Combining (8.36), (8.37) and (8.41) yields (8.35). Proof of (8.33). We have

({Q %J,(l) (α) > QJ,(l) (α) + ε} ∩ ΩJ,(l) ) P T k,T,n k,T  =P

Mn  1 

Mn

m=1

1{Φ  J,(l)

J,(l) k,T ,n,m >Qk,T (α)+ε}

Mn  1   ≤P 1{Φ  J,(l)

Mn

Mn − (αMn  − 1)

J,(l)  ∩ ΩT Mn



J,(l)

k,T ,n,m >Qk,T (α)+ε}

m=1

>

J,(l) 

− Υ(α, ε) > (1 − α) − Υ(α, ε) ∩ ΩT

(Φ with Υ(α, ε) = P k,T > Qk,T (α) + ε|X ). As the X -conditional distribution of J,(l)

J,(l)

J,(l)

J,(l)

Φk,T is almost surely continuous on ΩT for almost all ω ∈

J,(l) ΩT .

by Condition 8.7, we have Υ(α, ε) < 1−α

From (8.35) we then obtain

({Q %J,(l) > QJ,(l) (α) + ε} ∩ ΩJ,(l) ) → 0 P T k,T,n k,T because Mn 1  1 {Φ − Υ(α, ε) J,(l)  J,(l) Mn k,T ,n,m >Qk,T (α)+ε} m=1

J,(l)

converges on ΩT

in probability to zero by (8.35). Analogously we derive

({Q %J,(l) < QJ,(l) (α) − ε} ∩ ΩJ,(l) ) → 0 P T k,T,n k,T which together with (8.42) yields (8.33).

(8.42)

8.3 The Proofs

239

Proof of (8.34). Note that it holds J,(l) ck,T,n (α) =

√ %J,(l) nQk,T,n (1 − α)

(8.43)

nkV (l) (g4 , πn )T J,(l)

C,(l)

where the denominator in (8.43) converges to kCT > 0 on ΩT as shown in the proof of Theorem 8.3. Hence it remains to show that the numerator √ %J,(l) C,(l) nQk,T,n (1 − α) is oP (1) on ΩT . We have √

% nQ n,T (1 − α) ≤ J,(l)

√

Mn  n  % J,(l) Φk,T,n,m . αMn 

(8.44)

m=1

Further we get using (l)

(l)

(l)

P(|Δi,n (B (l) + C (l) )|p > β p |Ii,n |p |S) ≤

(l)

K|Ii,n |p + Kp |Ii,n |p/2 (l)

β p |Ii,n |p (l)

≤ Kp |Ii,n |p(1/2−) (l)

for p ≥ 1 where the process (Bt )t≥0 is defined as in (8.19), the elementary √ √ √ inequality a + b ≤ a + b which holds for all a, b ≥ 0, iterated expectations and twice the Cauchy-Schwarz inequality

E

√

≤

 

 % n|Φ k,T,n,m |1ΩC,(l) S

√

J,(l)



n

(l)

T

(l) E |Δi,n (B (l) + C (l) )|3 1{|Δ(l) (B (l) +C (l) )|>β|I (l) | } i,n

i:ti,n ≤T



≤

√

(l)

(l)

(l)

(l)

i,n

(l)

(l)

(l)

(l)

× (˜ σn (ti,n , −))2 ξ˜k,n,m,− (ti,n ) + (˜ σn (ti,n , +))2 ξ˜k,n,m,+ (ti,n )



n



1/2   S

(l)

E[(Δi,n (B (l) + C (l) ))12 |S]

(l)

i:ti,n ≤T (l)



(l)

× P(|Δi,n (B (l) + C (l) )|p > β p |Ii,n |p |S) (l)

(l)

(l)

(l)

× E[(˜ σn (ti,n , −))2 + (˜ σn (ti,n , +))2 |S]

≤ Kp n



|πn |T



(l) (Kq |Ii,n |12

1/2 

√ √ ≤ Kp n(|πn |T )1/2+p(1−2)/4 n

n|πn |T

(l) (l) + |Ii,n |6 )1/4 |Ii,n |p(1−2)/4 (4bn /bn )1/2

(l)

i:ti,n ≤T

K

1/4

 (l)

i:ti,n ≤T

(l)

|Ii,n |3/2 .

240

8 Testing for the Presence of Jumps

If we pick p such that p(1 − 2)/4 ≥ 1/2 the final bound vanishes as n → ∞ by Condition 8.7(i) and 8.2(ii); compare (8.30). Hence by Lemma 2.15 we obtain √ %J,(l) nQk,T,n (1 − α) = oP (1). Proof of Corollary 8.12. First observe that following the proof of Theorem 8.8 it P

holds n−1/2 Ak,T,n −→ 0 as n → 0 from which we obtain J,(l)

√

n−1 Ak,T,n J,(l)

nρ

n−1/2 Ak,T,n J,(l)

kV (l) (g4 , πn )T

1ΩJ,(l) = ρ T

kV (l) (g4 , πn )T

P

1ΩJ,(l) −→ 0.

(8.45)

T

Further using Corollary 2.38(a) and Theorem 2.41(a) we obtain P

Ak,T,n 1ΩC,(l) −→ (k2 Ck,T − kCT J,(l)

J,(l)

J,(l)

T

)1ΩC,(l) T

which yields J,(l)

ρ

k2 Ck,T − kCT J,(l)

Ak,T,n

P

knV (l) (g4 , πn )T

1ΩC,(l) −→ ρ

J,(l)

J,(l)

kCT

T

1ΩC,(l) .

(8.46)

T

The convergence (8.45) then yields together with Theorem 8.8, Proposition B.7(i) and Lemma B.5 the X -stable convergence √

 n(Φ k,T,n (ρ) − 1) = J,(l)

√  J,(l) n Φk,T,n − ρ



n−1 Ak,T,n J,(l)

kV (l) (g

4 , πn ) T

L−s

J,(l)

− 1 −→ Φk,T

J,(l)

on ΩT and hence (8.16) follows as in the proof of (8.14). From (8.46) and Theorem 8.3 we derive



n−1 Ak,T,n J,(l)

J,(l)

Φk,T,n − ρ

kV (l) (g4 , πn )T



k2 Ck,T − kCT J,(l)

P

− 1 1ΩJ,(l) −→ (1 − ρ) T

J,(l)

J,(l)

kCT

1ΩC,(l) T

which is almost surely larger than 0 by Condition 8.10(ii) and hence (8.17) follows as in the proof of (8.15).

9 Testing for the Presence of Common Jumps At the beginning of Chapter 8 we discussed that when modelling a univariate process in continuous time one has to decide whether to incorporate a jump component or not. The same problem occurs in a multivariate setting where multiple processes should be modelled at once. In the situation where a multivariate model with jumps has to be specified, not only the individual jump components have to be characterized but also the dependence structure of the individual jump components has to be modelled. Within the set of models where the jump components of different processes are dependent ”the easiest ones to tackle are those for which the various components do not jump together”([32], Section 1. Introduction). Further, in the situation where the processes model stock prices, common jumps of multiple processes play an important role in portfolio management. In fact, systemic jumps that affect the whole market are common jumps and have a significant effect on the portfolio value while on the other hand idiosyncratic jumps only play a minor role in a sufficiently diversified portfolio. Among other things, those two reasons motivate the need for formal tests that allow to decide whether common jumps in multiple processes do occur or not. In the following, we restrict ourselves to the bivariate situation as the question whether multiple processes jump together can be reduced to the question whether any two of these processes jump together. Like in Chapter 8 we are in general not able to answer the above question completely in the high-frequency framework because we only observe a single realization of the processes. Therefore we will develop tests based on asynchronous observations of these two processes that allow (l) to decide whether common jumps in the observed paths of two processes Xt (ω), t ∈ [0, T ], are present. So mathematically we are looking for a test based on the (l) observations X (l) (ω), i ∈ N0 , l = 1, 2, which allows to decide to which of the ti,n

following two subsets of Ω (1)

(2)

= {∃t ∈ [0, T ] : ΔXt ΔXt ΩCoJ T ΩnCoJ T

=

(ΩCoJ )c T

=

(1) (2) {ΔXt ΔXt

= 0}, = 0 ∀t ∈ [0, T ]}

ω belongs. In Section 9.1 we construct a test which works under the null hypothesis that no common jumps exist, i.e. ω ∈ ΩnCoJ , and in Section 9.2 we will look at T © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 O. Martin, High-Frequency Statistics with Asynchronous and Irregular Data, Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics, https://doi.org/10.1007/978-3-658-28418-3_9

242

9 Testing for the Presence of Common Jumps

two tests which work under the null hypothesis that common jumps do exist, i.e. ω ∈ ΩCoJ . All tests are based on the observation that by Theorem 2.3 the nonT normalized functional V (f (p1 ,p2 ) , πn )T with f (p1 ,p2 ) (x, y) = |x|p1 |y|p2 converges for p1 ∧ p2 ≥ 2 to a strictly positive value on ΩCoJ and to zero on ΩnCoJ . The T T upcoming tests can be understood as generalizations of two tests for common (l) jumps based on synchronous and equidistant observations ti,n = i/n from [32].

9.1 Null Hypothesis of Disjoint Jumps In this section we develop a statistical test which allows to decide whether common jumps are present in the paths of two stochastic processes under the null hypothesis that no common jumps are present.

9.1.1 Theoretical Results In the following we will work with V (f (p1 ,p2 ) , πn )T for p1 = p2 = 2. The function f (2,2) (x, y) = f(2,2) (x, y) = x2 y 2 is within the set of functions {f (p1 ,p2 ) |p1 , p2 ≥ 0} the one with the smallest exponents p1 , p2 such that V (f (p1 ,p2 ) , πn )T converges under no additional conditions on the observation scheme; compare Theorem 2.3. Further this function has due to the quadratic structure nice mathematical properties which will be exploited in the proofs. By Theorem 2.3 the functional V (f(2,2) , πn )T converges on ΩCoJ to a strictly T positive value and on ΩnCoJ to zero. Thereby using the statistic V (f(2,2) , πn )T we T are asymptotically able to differentiate between ΩCoJ and ΩnCoJ . However, the T T limit of V (f(2,2) , πn )T might also be very close to zero for ω ∈ ΩCoJ and the limit T itself might be difficult to interpret. Therefore we consider the following statistic ΨDisJ T,n = 

V (f(2,2) , πn )T V (1) (g

4 , πn )T V

(2) (g

4 , πn ) T

where we devide V (f(2,2) , πn )T by the geometric mean of V (l) (g4 , πn )T , l = 1, 2. J,(1)

The statistic ΨDisJ T,n is scale invariant and converges on the set ΩT

where both processes X (l) , l = 1, 2, have at least one jump in [0, T ] to ΨDisJ T

= 

B ∗ (f(2,2) )T B (1) (g4 )T B (2) (g4 )T

J,(2)

∩ ΩT



= 

(1) 2 (2) 2 s≤T (ΔXs ) (ΔXs ) . (1) 4 1/2   (2) 4 1/2 s≤T (ΔXs ) s≤T (ΔXs )

The random variable ΨDisJ can be interpreted as a correlation coefficient of T the squared jumps of X (1) and X (2) and thereby yields a measure for how big the proportion of common jumps is compared to the amount of total jumps of

9.1 Null Hypothesis of Disjoint Jumps

243

X (1) and X (2) . In fact by the Cauchy-Schwarz inequality for sums we find that ΨDisJ ∈ [0, 1] where ΨDisJ = 0 holds if and only if no common jumps exist and T T ΨnCoJ = 1 holds if and only if there exists some constant c > 0 such that T (1)

(2)

(ΔXs )2 = c(ΔXs )2 , s ∈ [0, T ], holds. Hence, the statistic ΨDisJ T,n does not only help to decide whether common jumps are present or not but its value is also of direct interest as it yields an indicator for how big a role common jumps play in the model. By using the statistic ΨDisJ T,n instead of V (f(2,2) , πn )T we are restricting ourselves to testing ω ∈ ΩCoJ against ω ∈ ΩDisJ where T T J,(1)

= ΩnCoJ ∩ ΩT ΩDisJ T T

J,(2)

∩ ΩT J,(1)

J,(2)

because the limit of ΨDisJ is only defined on ΩT ∩ ΩT . However, this is T no real limitation because using the tests from Chapter 8 we are able to decide J,(1) J,(2) in advance whether ω is in ΩT ∩ ΩT or not. Hence by combining the tests from Chapter 8 with the upcoming test we are also able to test ω ∈ ΩCoJ against T ω ∈ ΩnCoJ . T Theorem 9.1. Suppose Condition 1.3 is fulfilled. Then it holds P

DisJ 1ΩJ,(1) ∩ΩJ,(2) . ΨDisJ T,n 1ΩJ,(1) ∩ΩJ,(2) −→ ΨT T

T

T

(9.1)

T

As already discussed earlier, Theorem 9.1 yields that ΨDisJ T,n converges to 0 on CoJ the set ΩDisJ and to a strictly positive limit on Ω . So a natural test for the T T CoJ null ω ∈ ΩDisJ against ω ∈ Ω makes use of a critical region of the form T T



DisJ DisJ = ΦDisJ CT,n T,n > cT,n



(9.2)

DisJ (cT,n )n∈N .

DisJ for a suitable, possibly random sequence In order to choose cT,n such that the test has a certain level α we need knowledge of the asymptotic DisJ behaviour of ΦDisJ which will be developed in form of a central limit T,n on ΩT theorem in the following.

Analogously as in (8.7) we observe n(ΨCoJ T,n − 0) = 

nV (f(2,2) , πn )T V (1) (g4 , πn )T V (2) (g4 , πn )T

and hence it remains to find a central limit theorem for V (f(2,2) , πn )T which holds on ΩDisJ . To motivate the structure of the upcoming central limit theorem we T again use a toy example of the form Xttoy = σWt +

 s≤t

ΔN (q)s , t ≥ 0.

244

9 Testing for the Presence of Common Jumps

Then on the subset of ΩDisJ where any two jumps of N (q) are further apart than 2|πn |T it holds



nV (f(2,2) , πn )T = n

(1)

+

+



(1)

(2)

i,n

i,j:ti,n ∨tj,n ≤T





(ΔN (l) (q)S (l) )2 n

l=1,2

(l) p:Sq,p ≤T





(2)

(Δi,n C (1) )2 (Δj,n C (2) )2 1{I (1) ∩I (2) =∅}

q,p

(3−l)

(Δi,n

(3−l) i:ti,n ≤T

(9.3)

j,n

C toy )2 1{I (l)

(l) (l) in (Sq,p ),n

(3−l)

∩Ii,n

=∅}

(9.4) 2ΔN

(l)

(q)S (l) Δ q,p

l=1,2 p:S (l) ≤T q,p



×n

(l)) (l)

(l)

in (Sq,p ,n

(3−l)

(Δi,n

(3−l) i:ti,n ≤T

C

toy

C toy )2 1{I (l)

(l) (l) in (Sq,p ),n

(3−l)

∩Ii,n

(9.5)

=∅}

(l)

where Cttoy = σWt and Sq,p , p ∈ N, denotes an enumeration of the jump times of N (l) (q). Under appropriate assumptions on the observation scheme which yield (l) |Ii,n | = OP (n−1 ) we then obtain that (9.3) converges in probability by Theorem 2.22, that (9.4) converges stably in law using methods from Section 3.1 which are based on the stable convergence of the expressions n



(3−l)

(Δi,n

(3−l) i:ti,n ≤T

= σ (3−l) n

C toy )2 1{I (l)

 (3−l) i:ti,n ≤T

(l) (l) in (Sq,p ),n

(3−l)

(Δi,n

W

(3−l)

∩Ii,n

=∅}

(3−l) 2

) 1{I (l)

(l) (l) in (Sq,p ),n

(3−l)

∩Ii,n

=∅}

,

(9.6)

compare (3.23) for the definition of W , and that (9.5) is asymptotically negigible. Remark 9.2. Recall that we have already derived a central limit theorem for V (f(2,2) , πn )T in Section 3.1. In fact Theorem 3.6 yields √

L−s

n(V (f(2,2) , πn )T − B ∗ (f(2,2) )T ) −→ Φbiv T (f(2,2) ).

(1) However, Φbiv and X (2) and in T (f(2,2) ) only depends on common jumps of X

particular it holds Φbiv ≡ 0. But a central limit theorem where the T (f(2,2) )1ΩnCoJ T asymptotic distribution is singular is useless for statistical purposes. Hence we √ need to develop a different central limit theorem on ΩnCoJ . As nV (f(2,2) , πn )T T converges on ΩnCoJ to zero we need to multiply V (f(2,2) , πn )T by a factor which T √ increases faster in n than n. In the following, we will see that by using the factor n error terms that are usually dominated by terms involving common jumps contribute to the asymptotic variance. 

9.1 Null Hypothesis of Disjoint Jumps

245

Before we can state the central limit theorem (whose form we motivated by the computations in the toy example above) we need to introduce some notation. Using the process W defined in (3.23) we denote



(l)

ηn,− (s) =

(l)

(Δj,n W

(l) 2

) 1

(l)

j:Ij,n ≤T



(l)

ηn,+ (s) =

(l)

(Δj,n W

(l) 2

) 1

(l) j:Ij,n ≤T

(l)

(3−l) (l)

=∅∧jin (s) (3−l) in (s),n

Ij,n ∩I

Ij,n ∩I

,

,

and as in (3.6) we denote (l)

δn,− (s) = s − t

(l)

(l)

(l)

in (s)−1,n

, δn,+ (s) = t

(l) (l)

in (s),n

− s,

for l = 1, 2 and s ≥ 0; see Figure 9.1 for an illustration. Using the notation from (2.28) we then obtain the identity 

(l)

(Δj,n W

(l) 2

(l)

(l)

(l)

) = ηn,− (s) + (δn,− (s))1/2 ((W s − W

(l) (3−l) j:Ij,n ∩I (3−l)

=∅ in (s),n

(l)

+(δn,+ (s))1/2 ((W

(l)

(l)

(l)

τn,+ (s)

(3−l) n (s),n

(l)

(l)

(l)

− W s )/(δn,+ (s))1/2 )

for the sum of the squared increments of W with the observation interval Ii

(l) τn,− (s)

(l)

)/(δn,− (s))1/2 )

2

(l)

+ ηn,+ (s)

(l)

over intervals Ij,n which overlap

containing s. In the computations using our (3−l)

toy example in (9.6) this sum occured for s replaced with a jump time Sq,p

of

(l)

(3−l)

X . We additionally distinguish between increments of W before and after s to allow for different volatilities immediately before and after s due to a volatility jump at time s, and we write DisJ,(l)

Zn



(l)

(l)

(l)

(l)

(s) = nηn,− (s), nηn,+ (s), nδn,− (s), nδn,+ (s)

∗

to shorten notation. The following condition sums up the assumptions on the asymptotics of the sequence of observation schemes (πn )n∈N which are needed to obtain a central limit theorem. Condition 9.3. Suppose that Condition 1.3 is fulfilled and that the observation times are exogeneous. Further, n (i) the functions t → Gn 2,2 (t) and t → H0,0,4 (t), see (2.39) for their definition, converge pointwise on [0, T ] in probability to increasing continuous functions G2,2 , H0,0,4 : [0, ∞) → [0, ∞).

246

9 Testing for the Presence of Common Jumps (1)

(2)

(1)

Intervals for ηn,− (Sq,p )

(2)

(1)

δn,− (Sq,p )

(2)

δn,+ (Sq,p )

(1)

(2)

ηn,+ (Sq,p ) = 0

X (1) (2)

Sq,p X (2)

Δ

(2) (2)

(2)

in (Sq,p ),n DisJ,(1)

Figure 9.1: Illustration of Zn

X ≈ ΔX

(2) (2)

Sq,p

(2)

(2)

(Sq,p ) at a jump time Sq,p of X (2) .

(ii) For all P1 , P2 ∈ N and all bounded continuous functions g : RP1 +P2 → R (l) and hp : R4 → R, p = 1, . . . , Pl , l = 1, 2, the integral

(

 [0,T ]k1 +k2

g(x1 , . . . , xP1 , x 1 , . . . , x P2 )E

P1 

 (1)  DisJ,(1) Zn (xp )

hp

p=1

×

P2 

(2)  DisJ,(2)  hp Zn (xp )

) dx1 . . . dxP1 dx 1 . . . dx P2

p=1

converges to

 [0,T ]k1 +k2

×

g(x1 , . . . , xP1 , x 1 , . . . , x P2 )

P1   p=1

P2 



p=1

(2)

R

(1)

R

hp (y)ΓDisJ,(1) (xp , dy)

hp (y)ΓDisJ,(2) (x p , dy)dx1 . . . dxP1 dx 1 . . . dx P2

(9.7)

as n → ∞. Here, ΓDisJ,(l) (·, dy), l = 1, 2, are families of probability measures on [0, ∞)2 × (0, ∞)2 such that the first moments are uniformly bounded. Condition 9.3(i) is needed to obtain convergence of (9.3) in the motivating example which we will prove using Theorem 2.22. Further, part (ii) is a condition which in similar form already occured in Section 3.1 and which is necessary to obtain stable convergence of the term (9.4). Note that as we are going to derive a central DisJ,(l) (l) limit theorem restricted to ΩDisJ that the random variables Zn (Sq,p ) and T DisJ,(l )

Zn

(l )

(l)

(l )

(Sq,p ) for Sq,p = Sq,p or l =  l are asymptotically S-conditionally

9.1 Null Hypothesis of Disjoint Jumps

247

independent. This explains the factorization of the expectations with respect to ΓDisJ,(l) in (9.7). Here, the form is somewhat simpler compared to Condition 3.5(ii) DisJ,(1) because we do not have to keep track of the common distribution of Zn (s) DisJ,(2) and Zn (s). Next, we define the components occuring in the asymptotic variance of the upcoming central limit theorem. We denote



TDisJ = C

T 0

and

 TDisJ = D

(1) (2) (σs σs )2 dG2,2 (s) +

 

(1) 2

ΔXs



T 0

(1) (2)

2(ρs σs σs )2 dH0,0,4 (s)



(2) 2

RDisJ,(2) (s) + ΔXs



RDisJ,(1) (s) ,

s≤T

where Sp , p ∈ N, is an enumeration of the jump times of X. Here, RDisJ,(l) (s) is defined by (2)

(l)

(2)

(l)



(l)

(l)

(l)

(l)

(l)

(l)

RDisJ,(l) (s) = (σs− )2 η− (s) + σs− (δ− (s))1/2 U− (s) + σs (δ+ (s))1/2 U+ (s) + (σs )2 η+ (s), (l)

s ∈ [0, T ], (l)

(l)

2

l = 1, 2, (l)

where Z DisJ,(l) (s) = (η− (s), η+ (s), δ− (s), δ+ (s))∗ are random variables defined

) which are independent of each other  F, P on an extended probability space (Ω, and of F . Their distribution is given by Z P (l)

DisJ,(l)

(s)

(dy) = ΓDisJ,(l) (s, dy), s ∈ [0, T ].

(l)

Further, the U− (s), U+ (s) are i.i.d. N (0, 1) random variables which are inde-

)  F, P pendent of all previously introduced random variables and defined on (Ω, (l) (l) DisJ,(l) as well. The Z (s) and U− (s), U+ (s) are independent of each other and

independent for different values s. Note that here we chose Z DisJ,(1) (s) and DisJ,(1) DisJ,(2) Z DisJ,(2) (s) to be independent although Zn (s) and Zn (s) are in general not independent. However, as we derive a central limit theorem only on ΩDisJ T (1)

(2)

we have Δ(1) Xs = 0 or Δ(2) Xs = 0 for any s. Hence only one of RDisJ,(1) (s), RDisJ,(2) (s) and therefore also only one of Z DisJ,(1) (s), Z DisJ,(2) (s) enters the  TDisJ for any s ≥ 0. This property allows to conclude that D  TDisJ does not term D DisJ,(1) DisJ,(2) depend on the common distribution of Z (s), Z (s) at all which is why we may assume that they are independent for convenience. Theorem 9.4. If Condition 9.3 is fulfilled we have the X -stable convergence L−s

DisJ =  nΨDisJ T,n −→ ΦT

. on the set ΩDisJ T

TDisJ + D  TDisJ C B (1) (g4 )T B (2) (g4 )T

(9.8)

248

9 Testing for the Presence of Common Jumps

For the notion what it means for a sequence of random variables to converge X -stably in law on a subset of Ω compare the paragraph following Theorem 8.8. TDisJ in the limit ΦDisJ The term C corresponds to (9.3) in the motivating example T and is solely due to the continuous martingale part of X. Further, the term  TDisJ corresponds to (9.4) and contains the limit of cross terms of the continuous D martingale parts and of idiosyncratic jumps. To illustrate Condition 9.3 and Theorem 9.4 we are going to discuss both of them for the two prominent observation schemes of equidistant observation times and Poisson sampling which feature as the major examples throughout this book. Example 9.5. In the setting of equidistant and synchronous observation times (l) ti,n = i/n it holds |πn |T = n−1 and hence Condition 1.3 is trivially fulfilled. Furthermore, it holds t/n n Gn 2,2 (t) = H0,0,4 (t) = n

 

1/n

2

→t

i=1 (l)

(l)

which yields Condition 9.3(ii). We also have ηn,− (s) = ηn,+ (s) = 0 and hence (l) δn,− (s),

DisJ,(l) Zn (s)

(l)

for the convergence of it remains to consider δn,+ (s). Then Condition 9.3(iii) is fulfilled as shown in Example 3.3 and we obtain Z DisJ,(l) (s) ∼ (0, 0, κ(l) (s), 1 − κ(l) (s)) for i.i.d. U[0, 1]-distributed random variables κ(l) (s), s ∈ [0, t], l = 1, 2. Hence in the setting of equidistant and synchronous observation times the terms TDisJ and D  TDisJ in (9.8) are of the form C

TDisJ = C  TDisJ = D



T



0

  s≤T



(1) (2)

(1) (2)



(σs σs )2 + 2(ρs σs σs )2 ds,

2 (1) 2  (2) (2) (2) (2) (2) σs− (κ (s))1/2 U− (s) + σs (1 − κ(2) (s))1/2 U+ (s)

ΔXs

2  (2) 2  (1) (1) (1) (1) (1) σs− (κ (s))1/2 U− (s) + σs (1 − κ(1) (s))1/2 U+ (s) ,

+ ΔXs

 T in (3.12) and These terms are identical to the corresponding terms CT and D (3.14) of [32], and Theorem 9.4 becomes Theorem 4.1(a) of [32] in this specific setting.  Example 9.6. In the case of Poisson sampling, which is introduced in Definition 5.1, Condition 1.3 is satisfied by (5.2). Further, Condition 9.3(ii) is, as shown in Example 5.7, fulfilled with P

Gn 2,2 (t) −→

 2

λ1



+

2 t, λ2

P

n H0,0,4 (t) −→

2 t. λ1 + λ2

9.1 Null Hypothesis of Disjoint Jumps

249

Finally, part (iii) of Condition 9.3 is satisfied by Lemma 5.10 and the discussion following Lemma 5.10 yields that the distributions ΓDisJ,(l) (s, dy) do not depend on s and can be constructed as follows: By symmetry we focus on Z DisJ,(1) (s) (1) (1) (2) (2) only. Let Ek,− , Ek,+ ∼ Exp(λ1 ), k ∈ N, and E− , E+ ∼ Exp(λ2 ) be independent exponentially distributed random variables. Then, after rescaling, the lengths of the (l) observation intervals I (l) , l = 1, 2, containing s are by the memorylessness in (s),n

(1)

(1)

of the exponential distribution asymptotically distributed like E1,− + E1,+ and (2)

(2)

(1)

(1)

E− + E+ while the other intervals in η− (s) and η+ (s) are Exp(λ1 )-distributed. Then, if (Uk,− )k∈N , (Uk,+ )k∈N are i.i.d. N (0, 1) random variables, it is easy to deduce that L

(1)

(1)

(1)

(1)

Z DisJ,(1) (s) = (η− (s), η+ (s), E1,− , E1,+ )∗ holds, where we set (1)

∞ 

(1)

k=2 ∞ 

η− (s) = η+ (s) =

(1)

Ek,− (Uk,− )2 1{k−1 E (1) j=1

2 k,− β|I (l) | } R

DisJ,(3−l)

i,n

i,n

(l)

(ti,n )

i,n

with

%n,m R

DisJ,(l)







2 (l) (l) (l) (l) (l) (s) = σ ˜n (s, −) ηˆn,m,− (s) + σ ˜n (s, −)(δˆn,m,− (s))1/2 Un,m,− (s) (l)

(l)

(l)

+σ ˜n (s, +)(δˆn,m,+ (s))1/2 Un,m,+ (s)

2



(l)

2

(l)

+ σ ˜n (s, +) ηˆn,m,+ ,

].

252

9 Testing for the Presence of Common Jumps

s ∈ [0, T ], l = 1, 2. Hence to obtain X -conditionally independent and identicDisJ % T,n,m ally distributed random variables D , m = 1, . . . , Mn , whose F -conditional

 TDisJ we estimate distribution approximates the X -conditional distribution of D (l) jumps of X by unusually large increments, the spot volatility by the estimators (l) (l) DisJ,(l) %n,m σ ˜n (s, −), σ ˜n (s, +) introduced in Chapter 6 and Z DisJ,(l) (s) by Z (s). To DisJ  estimate quantiles of the X -conditional distribution of DT we define similarly as in (8.13) for α ∈ [0, 1] %DisJ % Q T,n (α) = Qα





DisJ  % T,n,m D m = 1, . . . , Mn



.

%DisJ We will see that Q T,n (α) consistently estimates the X -conditional α-quantile

 TDisJ on ΩDisJ QDisJ (α) of D . Here, QDisJ (α) ∈ [0, ∞] is defined as the (under T T T the upcoming condition unique) X -measurable [0, ∞]-valued random variable fulfilling  TDisJ ≤ QDisJ P( D (α)|X )(ω) = α, ω ∈ ΩDisJ , T T (α))(ω) = 0 for ω ∈ (ΩDisJ )c for completeness. and we set (QDisJ T T The following condition summarizes all additional assumptions we need in order to obtain an asymptotic test.

Condition 9.7. Suppose that Condition 9.3 is fulfilled and assume that the (1) (2) set {s ∈ [0, T ] : σs σs = 0} is almost surely a Lebesgue null set. The sequence P

(bn )n∈N fulfils bn → 0, |πn |T /bn −→ 0 and (Kn )n∈N , (Mn )n∈N are sequences of P

integers converging to infinity with Kn /n −→ 0. Additionally, (i) it holds

    DisJ,(lp )   P Z%n,1 P (sp ) ≤ xp , p = 1, . . . , P S     Z DisJ,(lp ) (sp ) ≤ xp , p = 1, . . . , P  > ε → 0 −P

(9.11)

as n → ∞, for all ε > 0, P ∈ N, x = (x1 , . . . , xP ) ∈ R4×P , lp ∈ {1, 2} and sp ∈ (0, T ), p = 1, . . . , P , with sp = sp for p = p . o semimartingale, i.e. a (ii) The volatility process σ is itself an R2×2 -valued Itˆ process of similar form as (1.1). (1) (2)

The assumption {s ∈ [0, T ] : σs σs = 0} ensures in particular that the volatility  TDisJ > 0 almost surely. Further (9.11) yields that does not vanish, which yields D the empirical distribution on the

%n,m Z

DisJ,(l)

(l)

(l)

(l)

(l)

(s) = (ˆ ηn,m,− (s), ηn,m,+ (s), δn,m,− (s), δn,m,+ (s)),

9.1 Null Hypothesis of Disjoint Jumps

253

m = 1, . . . , Mn , converges to the non-degenerate distribution of Z DisJ,(l) (s) which is essential for the bootstrap method to work. Part (ii) of Condition 9.7 is like in Chapters 4 and 8 needed to ensure the consistency of the estimators σ ˜ (l) (s, −), (l) σ ˜ (s, +) for the spot volatilites. Theorem 9.8. If Condition 9.7 is satisfied, the test defined in (9.2) with DisJ cT,n (α) =

%DisJ V − (4, f(2,2) , πn , (β, ))T + Q T,n (1 − α) n



V (1) (g4 , πn )T V (2) (g4 , πn )T

,

α ∈ [0, 1],

has asymptotic level α in the sense that we have

 DisJ   DisJ  ΨDisJ  P → α α ∈ [0, 1], T,n > cT,n (α) F

(9.12)

for all F DisJ ⊂ ΩDisJ with P(F DisJ ) > 0. Because of T

 CoJ   DisJ  ΨDisJ  P → 1 α ∈ (0, 1], T,n > cT,n (α) F

(9.13)

with P(F CoJ ) > 0 it is consistent as well. for all F CoJ ⊂ ΩCoJ T DisJ Although n appears in the definition of the critical value cT,n (α) it is not directly DisJ needed for the computation of cT,n (α) since it also occurs linearly in both

%DisJ V − (4, f(2,2) , πn , (β, ))T and Q T,n (1 − α). However, it enters indirectly through the choice of bn , Kn , Mn , but for which usually just a rough idea of the magnitude of n is needed.

Example 9.9. Condition 9.7(i) is fulfilled in the setting of synchronous equidistant %DisJ observation times discussed in Example 9.5 where our estimator Q T,n (1 − α) (d)

equals the estimator Zn (α) defined in (5.10) of [32] for Nn = Mn and any choice of Kn (not necessarily converging to infinity). Note that Kn → ∞ is needed to estimate the distribution of the observation intervals around some time s from the realization of multiple asymptotically similar and independent observation intervals around times s close to s. Hence, because here all observation intervals are identical the choice Kn = 1, n ∈ N, is sufficient.  Example 9.10. Condition 9.7 is fulfilled in the Poisson setting discussed in P

Example 9.6 if we choose (bn )n∈N appropriately. Indeed |πn |T /bn −→ 0 follows from (5.2) for every bn = O(n−γ ) with γ ∈ (0, 1) and that (9.11) holds is shown by Lemma 5.12. 

9.1.2 Simulation Results As in Section 8.2 we conduct a simulation study to verify the finite sample properties of the introduced methods. Our benchmark model is the one from Section 6 of [32],

254

9 Testing for the Presence of Common Jumps

Table 9.1: Parameter settings for the simulation. Case I-j II-j III-j I-m II-m III-m I-d0 II-d0 III-d0 I-d1 II-d1 III-d1

ρ 0.0 0.0 0.0 0.5 0.5 0.5 0.0 0.0 0.0 1.0 1.0 1.0

α1 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

κ1

l1

h1

1 5 25 1 5 25 1 5 25

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

0.7484 0.3187 0.1238 0.7484 0.3187 0.1238 0.7484 0.3187 0.1238

α2 0.00 0.00 0.00 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

Parameters κ2 l1

1 5 25 1 5 25 1 5 25

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

h1

α3 0.01 0.01 0.01 0.01 0.01 0.01

0.7484 0.3187 0.1238 0.7484 0.3187 0.1238 0.7484 0.3187 0.1238

κ3 1 5 25 1 5 25

l3 0.05 0.05 0.05 0.05 0.05 0.05

h3 0.7484 0.3187 0.1238 0.7484 0.3187 0.1238

because by using the same configuration as in their paper we are able to directly compare our approach to the case of equidistant and synchronous observations. Further the model can be understood as a natural extension of (8.18) to the bivariate setting. The model for X is given by (1) dXt (2) dXt

=

(1) (1) Xt σ1 dWt

=

(2) (2) Xt σ2 dWt

 + α1 + α2

R R

(1) Xt− x1 μ1 (dt, dx1 ) + α3 (2) Xt− x2 μ2 (dt, dx2 ) + α3



(1)

R R

Xt− x3 μ3 (dt, dx3 ), (2)

Xt− x3 μ3 (dt, dx3 ),

where [W (1) , W (2) ]t = ρt and the Poisson measures μi are independent of each other and have predictable compensators νi of the form νi (dt, dxi ) = κi

1[−hi ,−li ]∪[li ,hi ] (xi ) 2(hi − li )

dtdxi

where 0 < li < hi for i = 1, 2, 3, and the initial values are X0 = (1, 1)∗ . We consider the same twelve parameter settings which were discussed in [32] of which six allow for common jumps and six do not. In the case where common jumps are possible, we only use the simulated paths which contain common jumps. For the parameters we set σ12 = σ22 = 8 × 10−5 in all scenarios and choose the parameters for the Poisson measures such that the contribution of the jumps to the total variation remains approximately constant and matches estimations from real financial data; see [25]. The parameter settings are summarized in Table 9.1; compare Table 1 in [32]. To model the observation times we use the Poisson setting discussed in Examples 9.6 and 9.10 for λ1 = λ2 = 1 and set T = 1 which amounts on average to n observations of each X (1) and X (2) in the interval [0, T ]. We choose n = 100, n = 400 and n = 1,600 for the simulation. In a trading day of 6.5 hours this

9.1 Null Hypothesis of Disjoint Jumps

1.0

1.0

1.0

1.0

1.0 0.5 1.0 0.5

1.0

1.0

1.0

1.0 0.5

1.0 0.0 0.5

0.5

III-d1

0.5

1.0 0.0 0.0

0.0

III-d0

0.5

1.0 0.5

1.0

1.0

0.0 0.0

III-m

0.0

0.5

0.5

0.5

1.0 0.5

0.0

II-d1

0.0 0.0

III-j

0.0

1.0

0.5

1.0 0.5 0.5

0.5

II-d0

0.0

0.5 0.0 0.0

0.0

II-m

1.0

II-j

0.5

0.0

0.5 0.0

0.0

0.5

0.0

0.5 0.0

0.5 0.0 0.0

I-d1

1.0

I-d0

1.0

I-m

1.0

I-j

255

0.0

0.5

1.0

0.0

0.5

1.0

Figure 9.3: Empirical rejection curves from the Monte Carlo simulation for the test derived from Theorem 9.8. The dotted line represents the results for n = 100, the dashed line for n = 400 and the solid line for n = 1,600. In each case N = 10,000 paths were simulated. corresponds to observing X (1) and X (2) on average every 4 minutes, every 1 minute and every 15 seconds. We set β = 0.03 and  = 0.49 for all occuring √ (l) truncations. We use bn = 1/ n for the local interval in the estimation of σs and (l) %n,m Kn = ln(n), Mn = n in the simulation of the Z (s). For an explanation of this choice of parameters see Section 8.2. In Figure 9.3 we display the results from the simulation. The plots are constructed DisJ as follows: First for different values of α the critical values cT,n (α) are simulated according to Theorem 9.8. Then we plot the observed rejection frequencies against α. The six plots on the left show the results for the cases where the alternative of common jumps is true. In the cases I-j, II-j and III-j there exist only joint jumps and the Brownian motions W (1) and W (2) are uncorrelated. In the cases

256

9 Testing for the Presence of Common Jumps

I-m, II-m and III-m we have a mixed model which allows for disjoint and joint jumps and also the Brownian motions are positively correlated. The prefixes I, II and III indicate an increasing number of jumps present in the observed paths. Since our choice of parameters is such that the overall contribution of the jumps to the quadratic variation is roughly the same in all parameter settings, this corresponds to a decreasing size of the jumps. Hence in the cases I-* we have few big jumps while in the cases III-* we have many small jumps. We observe that the test has very good power against the alternative of common jumps. The power is greater for small n if there are less and bigger jumps as can be seen from the dotted lines for the cases I-j and I-m because the bigger jumps are detected more easily. On the other hand the power is greater for large n if there are more and smaller jumps which can be seen from the solid lines for III-j and III-m, because then it is more probable that at least one of the common jumps is detected and one small detected common jump is sufficient for rejecting the null. The six plots on the right in Figure 9.3 show the results for the cases where the null hypothesis is true. While in the cases *-d0 the Brownian motions W (1) and W (2) are uncorrelated, the Brownian motions are perfectly correlated in the cases *-d1. The prefixes I, II and III stand for an increasing number and a decreasing size of the jumps as in the first six cases. Under the null of disjoint jumps, we see that the observed rejection frequencies match the predicted asymptotic rejection probabilities from Theorem 9.8 very well in all six cases. There are slight deviations for a higher number of jumps. This is due to the fact that disjoint jumps whenever they lie close together, sometimes cannot be distinguished based on the observations which leads to over-rejection under the null hypothesis. In the cases *-d1 where the Brownian motions are perfectly correlated the rejection frequencies are systematically too high for large n. The results are worse than in the cases *-d0. In general, the results from the Monte Carlo are very similar to the results displayed in Figure 5 (note that the values for n there are 100, 1,600 and 25,600) from [32]. On a closer look, we observe that the power of our test in the asynchronous setting is slightly worse than the power of the test in the equidistant and synchronous setting while under the null hypothesis the rejection levels match the asymptotic levels more closely than in [32]. The loss in power is most pronounced for the smallest observation frequency n = 100 and in the range of at most a few percentage points for the more relevant frequency n = 1,600. Our results in the cases *-d1 are better than in [32] because the effect of a high correlation in the Brownian motions has less influence on the test statistic due to the asynchronicity. All in all, we conclude that there is no significant drawback of working with asynchronous observations instead of synchronous observations when testing for disjoint jumps in a bivariate process.

9.1 Null Hypothesis of Disjoint Jumps

257

9.1.3 The Proofs Proof of Theorem 9.1. The convergence (9.1) follows from Theorem 2.3, Corollary 2.2 and the continuous mapping theorem for convergence in probability. Proof of Theorem 9.4. Step 1. Using Condition 9.3(ii) and Theorem 2.22 we obtain



n

(1)

(1)

P

(2)

TDisJ . (9.14) (Δi,n C (1) )2 (Δj,n C (2) )2 1{I (1) ∩I (2) =∅} −→ C

(2)

i,n

i,j:ti,n ∨tj,n ≤T

j,n

This has been specifically computed in (2.43) in Example 2.23.  q) introduced in Step 1 Step 2. Recall the discretized versions σ ˜ (r, q) and C(r, in the proof of Theorem 3.6 and define 

 TDisJ (q, r) = n D





(l)

Δi,n N (l) (q)

2 

i,n

l=1,2 i,j:t(l) ∨t(3−l) ≤T i,n

2

(3−l)

(3−l) (r, q) 1 (l) (3−l) Δj,n C . {I ∩I

=∅} i,n

j,n

(l)

Let Sq,p , p ∈ N, be an enumeration of the jump times of N (l) (q) and denote by (l )

(l )

where two different jump times Sq,p1 1 = Sq,p2 2 are Ω(n, q, r) the subset of ΩDisJ T (l)

further apart than 4|πn |T and the jump times Sq,p are further away than 2|πn |T from the discontinuities k/2r of σ(r). On this set we get DisJ  T,n D (q, r)1Ω(n,q,r) =

  

ΔN (l) (q)S (l)

2

q,p

l=1,2 p:S (l)

(3−l)

n R

(l)

(Sq,p , r, q)1Ω(n,q,r)

q,p

where (l)

(l)

(l)

(l)

(l)

(l)

n (s, r, q) = (˜ R σs− (r, q))2 ηn,− (s) + σ ˜s− (r, q)(δn,− (s))1/2 Un,− (s) (l)

(l)

(l)

+σ ˜s (r, q)(δn,+ (s))1/2 Un,+ (s) (l)

2

(l)

(l)

+ (˜ σs (r, q))2 ηn,+ (s),

s ∈ [0, T ],

l = 1, 2,

(l)

with Un,− (s), Un,+ (s) defined as in (3.40). Comparing the proof of (3.25) we find that Condition 9.3(ii) yields the X -stable convergence



(1)

(2)

n (Sq,p , r, q) R





(2) Sq,p ≤T

L−s

−→

(2) (1) n , R (Sq,p , r, q)



(2)





(1) (Sq,p , r, q) R

(2) Sq,p ≤T



(2) Sq,p ≤T



(1) (2) (Sq,p , R , r, q)



 (2) Sq,p ≤T

where (l)

(l)

(l)

(l)

(l)

(l) (s, r, q) = (˜ R σs− (r, q))2 η− (s) + σ ˜s− (r, q)(δ− (s))1/2 U− (s) (l)

(l)

(l)

+σ ˜s (r, q)(δ+ (s))1/2 U+ (s)

2

(l)

(l)

+ (˜ σs (r, q))2 η+ (s),

s ∈ [0, T ],

l = 1, 2.

258

9 Testing for the Presence of Common Jumps

Using this stable convergence, Lemma B.3 and the continuous mapping theorem for stable convergence stated in Lemma B.5 we then obtain

  

ΔN (l) (q)S (l)

2

q,p

l=1,2 p:S (l)

(3−l)

n R

q,p

L−s

 TDisJ (q, r) := −→ D

(l)

(Sq,p , r, q)

  

ΔN (l) (q)S (l)

2

q,p

l=1,2 p:S (l)

(l)

(3−l) (Sq,p , r, q). R

q,p

Because of P(Ω(n, q, r)) → 1 as n → ∞ for any q, r > 0 this convergence yields L−s

DisJ  T,n  TDisJ (q, r) D (q, r) −→ D

(9.15)

for any q, r > 0. A more detailed proof of (9.15) can be found in the proof of Proposition A.3 in [36]. In fact the proof of Proposition A.3 in [36] is up to different notation almost identical to the proof of (3.25) which was presented in full detail in Chapter 3. Step 3. Next we show lim lim sup lim sup P

q→∞ r→∞

n→∞

  

 nV (f(2,2) , πn )T − R(n, q, r)T  > ε ∩ ΩnCoJ =0 T (9.16)

for all ε > 0 where R(n, q, r)T = n

 (1)



(1)

Δi,n C (1)

2 

(2)

Δj,n C (2)

2



(1)

+ Δi,n N (1) (q)

2 

(2)

(2) (r, q) Δj,n C

2

(2)

i,j:ti,n ∨tj,n ≤T



(1)

(1) (r, q) + Δi,n C

2 

(2)

Δj,n N (2) (q)

2 

1{I (1) ∩I (2) =∅} . i,n

j,n

As the proof of (9.16) is rather technical and more involved compared to the other steps it will be given later. Step 4. Because σ may assumed to be bounded and because the laws of ΓDisJ (·, dy) have uniformly bounded first moments we obtain similarly as in the proof of (3.26) lim lim sup P

q→∞ r→∞

   DisJ

 D  TDisJ (q, r) > ε ∩ ΩDisJ T −D = 0. T

(9.17)

Step 5. Using (9.14) and (9.15) together with Lemma B.3 and the continuous mapping theorem for stable convergence in law which is stated in Lemma B.5 we obtain L−s

TDisJ + D  TDisJ (q, r). R(n, q, r)T −→ C

(9.18)

9.1 Null Hypothesis of Disjoint Jumps

259

Further, combining (9.16), (9.17) and (9.18) and using Lemma (B.6) we obtain L−s

TDisJ + D  TDisJ nV (f(2,2) , πn )T −→ C P

which in combination with V (l) (g4 , πn )T −→ B (l) (g4 )T due to Corollary 2.2 yields (9.8). For the proof of (9.16) we need the following Lemma. Lemma 9.11. Let Condition 1.3 be satisfied, the processes σt , Γt be bounded and the observation scheme be exogenous. Then there exists a constant K which is independent of (i, j) such that

  (1)  (2) 

 (l) 2  (3−l) 2  E Δi,n M (q) Δj,n M (q ) 1ΩDisJ σ(Ft(1) ∧t(2) , S) ≤ Keq eq Ii,n Ij,n . T

i,n

j,n

(9.19) Proof. Arguing similarly as in the proof of Lemma 3.9 it is sufficient to prove (1) (2) (9.19) for Ii,n = Ij,n . The claim now follows from (8.17) in [32] which is basically (l)

(3−l)

(9.19) for Ii,n = Ij,n proof.

. The generalization to q =  q here does not complicate the

Proof of (9.16). Since γ is bounded by Condition 1.3 we can write X = X0 + B(q ) + C + M (q )

(9.20)

on [0, T ] for some positive number q (not necessarily an integer) which yields N (q) = B(q ) − B(q) + M (q ) − M (q).

(9.21)

First we observe that it holds lim lim sup P(|R(n, q)T − R(n, q, r)T | > ε) = 0

r→∞ n→∞

(9.22)

for any ε > 0 and any q > 0 where   (1) (1) 2  (2) (2) 2  (1) (1) 2  (2) (2) 2 R(n, q)T = n Δi,n C Δj,n C + Δi,n N (q) Δj,n C (1)

(2)

i,j:ti,n ∨tj,n ≤T



(1)

+ Δi,n C (1)

2 

(2)

Δj,n N (2) (q)

2 

1{I (1) ∩I (2) =∅} i,n

j,n

using similar arguments as for (3.34) in the proof of (3.24). Next, an application of inequality (2.46) with (l)

(l)

xl = Δi,n C (l) + Δi,n N (l) (q),

(l)

(l)

yl = Δi,n B (l) (q) + Δi,n M (l) (q)

260

9 Testing for the Presence of Common Jumps

yields

 q)T )1Ω(n,q) (nV (f(2,2) , πn )T − R(n,  q)T ≤ θ(ε)R(n, 

+ Kε



l=1,2 i,j:t(l) ∨t(3−l) ≤T

×



i,n

 (l) 2 1{I (l) ∩I (3−l) =∅} Δi,n (B (l) (q) + M (l) (q)) i,n

j,n

j,n

(3−l)

Δj,n (B (3−l) (q) + M (3−l) (q))

2



(3−l)

+ Δj,n (C (3−l) + N (3−l) (q))

2 

where



 q)T = R(n,

(1)



(1)

Δi,n (C (1) + N (1) (q))

2 

(2)

Δj,n (C (2) + N (2) (q))

2

(2)

i,j:ti,n ∨tj,n ≤T

× 1{I (1) ∩I (2) =∅} . i,n

j,n

It holds



 q)T ≤ 4 R(n,

 

(1) (2) ∨ti ,n ≤T 1 ,n 2

i1 ,i2 :ti

(l)

(l)

(Δil ,n C (l) )2 + (Δil ,n N (l) (q))2



l=1,2

× 1{I (1)

(2) i1 ,n ∩Ii2 ,n =∅}

 q)T is bounded in probability and hence by (9.14), (9.15) and (9.22) the term R(n,  for n, q → ∞ which yields that θ(ε)R(n, q)T vanishes as first n → ∞, then q → ∞ and finally ε → 0. Further, we also obtain for l = 1, 2 using (9.21), Lemma 1.4, Lemma 3.9 and Lemma 9.11   (l) (l) 2  (3−l) (3−l) 2  E Kε n Δi,n B (q) + M (l) (q) Δj,n (B (q) + M (3−l) (q) (l)

(3−l)

i,j:ti,n ∨tj,n

+



≤T

(3−l) Δj,n C (3−l)



≤ Kε n (l)

(3−l)

i,j:ti,n ∨tj,n

≤T

  2  (3−l) + Δj,n N (3−l) (q) 1{I (l) ∩I (3−l) =∅} 1ΩnCoJ S T i,n i,n   (l)   (l)  1{I (l) ∩I (3−l) =∅} Kq Ii,n  + Keq Ii,n  i,n

i,n

 (3−l)    (3−l)   (3−l)  × Kq Ij,n  + Keq + 2K + 8(Kq + Kq )Ij,n  + 8K(eq + eq ) Ij,n     ≤ Kε Kq |πn |T + Keq Kq |πn |T + Keq + K Gn 2,2 (T ),

where the latter bound converges to zero for n → ∞ and then q → ∞ for any ε > 0. Hence we obtain

 q)T | > δ} ∩ ΩnCoJ lim lim sup lim sup P({|nV (f(2,2) , πn )T − R(n, ) = 0 (9.23) T

ε→0 q→∞

n→∞

9.1 Null Hypothesis of Disjoint Jumps

261

for any δ > 0 and it remains to show that, restricted to the set ΩnCoJ , the quantity T  q)T − R(n, q)T vanishes as first n → ∞ and then q → ∞. To this end note R(n,  q)T − R(n, q)T is equal to that R(n, n







(l)

(l)

(3−l)

(3−l)

(l)

(l)

(Δi,n C (l) )2 + 2(Δi,n C (l) )(Δi,n N (l) (q)) + (Δi,n N (l) (q))2



l=1,2 i,j:t(l) ∨t(3−l) ≤T i,n

j,n





× 2 (Δj,n C (3−l) )(Δj,n N (3−l) (q)) 1{I (l) ∩I (3−l) =∅} . i,n

j,n

(l)

(3−l)

Here, the sum over terms containing the product (Δi,n N (l) (q))(Δj,n N (3−l) (q)) converges to zero because we are on ΩnCoJ . For the remaining terms we obtain T



n

(l)

(l) (3−l) i,j:ti,n ∨tj,n ≤T

≤



(3−l)

i,n

(3−l) j:tj,n ≤T





(3−l)

sup

(3−l)

(Δi,n C (l) )2 (Δj,n C (3−l) )(Δj,n N (3−l) (q))1{I (l) ∩I (3−l) =∅}

Δj,n C (3−l) n (l)

(l)

j,n

(3−l)

(Δi,n C (l) )2 (Δj,n N (3−l) (q))

(3−l)

i,j:ti,n ∨tj,n

≤T

× 1{I (l) ∩I (3−l) =∅} i,n

j,n

where, restricted to ΩnCoJ , the right-hand side tends to zero as n → ∞ for all T q > 0 because the supremum vanishes since C is continuous and because the sum converges stably in law on ΩnCoJ to T

 (3−l)

Sq,p

(3−l)

ΔN (3−l) (q)S (3−l) R(l) (Sq,p

),

q,p

≤T

(3−l)

where Sq,p , p ∈ N, denotes an enumeration of the jump times of N (3−l) (q). The stable convergence can be proven similarly as (9.15) and follows from Condition 9.3(ii). Proof of (9.10). Looking at the proof of (9.23), it is enough to show that n

 (1)



(1)

(1)

Δi,n C (1) + Δi,n N (1) (q)

2 

(2)

(2)

Δj,n C (2) + Δj,n N (2) (q)

2

(2)

i,j:ti,n ∨tj,n ≤T

× 1{|Δ(1) X (l) |≤β|I (1) | ∧|Δ(2) X (2) |≤β|I (2) | } 1{I (1) ∩I (2) =∅} i,n

i,n

T . converges on the set ΩDisJ to C T

j,n

j,n

i,n

j,n

(9.24)

262

9 Testing for the Presence of Common Jumps (l)

We first deal with the terms in (9.24) involving big jumps. Let Sq,p , p ∈ N, denote tan enumeration of the jump times of N (l) (q). Then it holds

   n





(l)

l=1,2 i,j:t(l) ∨t(3−l) ≤T

×



i,n

(l)

Δi,n N (l) (q) Δi,n (C (l) + N (l) (q)

j,n

(3−l) Δj,n (C (3−l)

+ N (3−l) (q))



2  

× 1{|Δ(l) X (l) |≤β|I (l) | ∧|Δ(3−l) X (3−l) |≤β|I (3−l) | } 1{I (l) ∩I (3−l) =∅} 1Ω(n,q,T ) ≤n



i,n

i,n



 ΔN (l) (q)

l=1,2 p:S (l) ≤T

 (l) × Δ (l)

j,n

  (l) 1

Sq,p

q,p

(l)

in (Sq,p ),n

j,n

{|Δ

j,n

(l) (l) X (l) |≤β|I (l) (l) | } (l) (l) in (Sq,p ),n in (Sq,p ),n





(C (l) + N (l) (q))

i,n

(3−l)

j:tj,n



(3−l)

(3−l)

Δj,n C (3−l) + Δj,n N (3−l) (q)

2

≤T

(9.25) where Ω(n, q, T ) denotes the set where any two different jumps of N (q) in [0, T ] are further apart than 2|πn |T . (9.25) converges in probability to zero as n → ∞ because of |Δ

P

(l) (l)

(l)

in (Sq,p ),n

X X| −→ |ΔN (l) (q)S (l) | > 0, q,p

|I

P

(l) (l)

(l)

in (Sq,p ),n

X | −→ 0

where PX denotes convergence in X -conditional probabilities. P(Ω(n, q, T )) → 1 as n → ∞ then yields that the terms involving big jumps in (9.24) vanish asymptotically. Hence only the terms involving squared increments of C (1) , C (2) contribute in the limit. Using (9.14) it is sufficient to show



T,n = n L

(1)



(1)

Δi,n C (1)

2 

(2)

Δj,n C (2)

2

(2)

i,j:ti,n ∨tj,n ≤T P

× 1{|Δ(1) X (1) |>β|I (1) | ∨|Δ(2) X (2) |>β|I (2) | } 1{I (1) ∩I (2) =∅} −→ 0. i,n

i,n

j,n

j,n

i,n

(9.26)

j,n

To this end note that the conditional Markov inequality plus an application of inequality (1.12) give

   (l) (l) (l) P |Δi,n X (l) | > β|Ii,n | S ≤ K|Ii,n |1−2 .

(9.27)

9.1 Null Hypothesis of Disjoint Jumps

263

Then using 1A∨B ≤ 1A + 1B for events A, B ∈ F and the Cauchy-Schwarz inequality, as well as the inequalities (1.9) and (9.27), we get

 

T,n S E L



≤ Kn

(1)

(2)

|Ii1 ,n ||Ii2 ,n |

(1) (2) ∨ti ,n ≤T 2 l ,n



(l)

|Iil ,n |(1−2)/2 1{I (1) ∩I (2) =∅} i,n

l=1,2

i1 ,i2 :ti

j,n

≤ K(|πn |T )(1−2)/2 Gn 2,2 (T ),

(9.28)

which tends to zero by Condition 1.3 and Condition 9.3(i). This estimate proves (9.26). To prove (9.12) in Theorem 9.8 we first prove a proposition similarly as in Chapters 4, 7 and 8. Proposition 9.12. Suppose that Condition 9.7 is satisfied. Then

 P

Mn   1   1{D  DisJ

Mn

m=1

 



T ,n,m ≤Υ}





 D  TDisJ ≤ ΥX  > ε ∩ ΩDisJ −P → 0 (9.29) T

as n → ∞ for any X -measurable random variable Υ and all ε > 0. Proof. Step 1. Denote by Sq,p , p ∈ N, an increasing sequence of stopping times which exhausts the jump times of N (q). Recall that on ΩDisJ only one component T of X jumps at Sq,p , and we use lp as the index of the component involving the p-th jump. Therefore, setting



An,p = Δ

(lp )

(l )

in p (Sq,p ),n

(3−lp )



σ ˜n

X (lp ) 1

{|Δ

(3−lp )

(Sq,p , −), σ ˜n

(l )

(lp ) (l ) X (lp ) |>β|I (lp ) | } (lp ) p in (Sq,p ),n in (Sq,p ),n

(3−l )

(Sq,p , +)

,



(3−l ) 

p DisJ p p %n,m %n,m Ap = ΔXSq,p , σSq,p − , σSq,p p , Z =Z (Sq,p ), Z p = Z DisJ (Sq,p )

and defining ϕ via ϕ((Ap , Y (lp ) (Sq,p ))p=1,...,P ) =

P 

(l )

p (ΔXSq,p )2 R(3−lp ) (Sq,p )1{Sq,p ≤T } ,

p=1

Lemma 4.9 proves

 P

Mn     1  

  Y (P ) ≤ ΥX  > ε ∩ ΩDisJ  1{Y (P,n,m)≤Υ} − P → 0 (9.30) T

Mn

m=1

264

9 Testing for the Presence of Common Jumps

where we used the notation P  

Y (P, n, m) =

Δ

p=1

(lp )

(l )

in p (Sq,p ),n

X (lp )

2

1

{|Δ

(lp ) (l ) X (lp ) |>β|I (lp ) | } (lp ) p in (Sq,p ),n in (Sq,p ),n

(3−l )

%n,m p (Sq,p )1{S ≤T } , ×R q,p Y (P ) =

P  

(l ) 2

p ΔXSq,p

R(3−lp ) (Sq,p )1{Sq,p ≤T } .

p=1

Step 2. Next, we prove lim lim sup

P →∞ n→∞

Mn   1   DisJ  % T,n,m P Y (P, n, m) − D >ε =0 Mn

(9.31)

m=1

for all ε > 0. Denote by Ω(P, q, n) the set on which there are at most P jumps of N (q) in [0, T ] and two different jumps of N (q) are further apart than |πn |T . Obviously, P(Ω(P, q, n)) → 1 for P, n → ∞ and any q > 0. On the set Ω(P, q, n) we have

  DisJ  Y (P, n, m) − D % T,n,m 1ΩDisJ 1Ω(P,q,n) T    (l) ≤

(l)

(l)

Δi,n B(q) + Δi,n C + Δi,n M (q)

l=1,2 t(l) ≤T,p:S i,n

(l) q,p ∈Ii,n

(3−l)

≤2



(l)

%n,m (t ) × 1{|Δ(l) X (l) |>β|I (l) | } R i,n i,n

 

l=1,2 t(l) ≤T

i,n

(l)

(l)

(l)

2

Δi,n B(q) + Δi,n C + Δi,n M (q) 1{|Δ(l) X (l) |>β|I (l) | } i,n

i,n

×

2

1



bn

(3−l)

j:Ij,n

(l)



(3−l)

Δj,n X

2 

i,n

(3−l)

(l)

ηˆn,m (ti,n )

(9.32)

(l)

⊂[ti,n −bn ,ti,n +bn ]

where we used the notation (3−l)

(l)

ηˆn,m (ti,n ) (3−l) (3−l) (3−l) (3−l) (3−l) (3−l) = ηˆn,m,− (s) + δˆn,m,− (s)Un,m,− (s)2 + δˆn,m,+ (s)Un,m,+ (s)2 + ηˆn,m,+ (s).

We first consider the increments over the overlapping observation intervals in the right-hand side of (9.32). The F -conditional mean of their sum is bounded by 3|πn |T  n bn





(l)

(l)

l=1,2 i,j:t(l) ∨t(3−l) ≤T i,n

j,n



(3−l)

× 1{|Δ(l) X (l) |>β|I (l) | } Δj,n X i,n

(l)

Δi,n B(q) + Δi,n C + Δi,n M (q)

i,n

2

2

1{I (l) ∩I (3−l) =∅} , i,n

j,n

(9.33)

9.1 Null Hypothesis of Disjoint Jumps

265

M1n (s)

s

M2n (s) (l)

Figure 9.4: Illustration for the variables Mn (s), l = 1, 2.



(l)

since with Mn (s) =

(l)

(l)

i:ti,n ≤T

|Ii,n |1{I (l) ∩I (3−l)



(3−l) (l)   E ηˆn,m (ti,n )F = n

 (3−l) (l) I ∩I k1 ,n

k1 ∈Z,|k2 |≤Kn

×





Kn 

≤

i+k2

j1 ,n

Kn 

 (l)  I 

k2 =−Kn

i+k2 ,n

nMn

k=−Kn ,...,Kn

we get

 

i+j2 ,n

 (l) I

i+j2 ,n

j2 =−Kn (3−l)

sup

=∅}

 (3−l) (l) I ∩I

j1 ∈Z,|j2 |≤Kn

=n

(3−l) in (s),n

i,n

−1 (3−l) (l)  Mn (t

i+k2 ,n )

−1 (3−l) (l)  Mn (t

i+k2 ,n )

(l)

(ti+k,n ) ≤ 3n|πn |T .

(9.34)

Because of Theorem 9.4 the sum in (9.33) is of order n−1 on ΩDisJ , while T P

|πn |T /bn → 0 for n → ∞ by Condition 9.7. Hence, (9.33) vanishes as n → ∞ for any q > 0. Next, we deal with the increments over non-overlapping observation intervals in the right-hand side of (9.32). An upper bound is obtained by taking iterated S-conditional expectations using Lemma 1.4, the Cauchy-Schwarz inequality as in (9.28) and (9.34), and it is given by



 

(l)

(l)

(l)

Kq |Ii,n |2 + K|Ii,n |1+(1−2)/2 + Keq |Ii,n |

 2Kbn

l=1,2 t(l) ≤T

bn

i,n

×n



Kn   (l)  I 

Kn 

k=−Kn

 (l) I

i+k ,n

i+k

k =−Kn

≤ K Kq |πn |T + (|πn |T )(1−2)/2 + eq

−1 (3−l) (l)  Mn (t

 

i+k,n )

  (l)  I  i,n

l=1,2 t(l) ≤T i,n

266

9 Testing for the Presence of Common Jumps Kn   (l)  I 

×n

Kn 

 (l) I

i+k ,n

i+k

−1 (3−l) (l)  Mn (t

i+k,n ).

k =−Kn

k=−Kn

Now (9.31) follows from Condition 9.3(i) because of n





l=1,2 t(l) ≤T i,n

=n

Kn   (l)  I 

(l)

|Ii,n |



Kn 

i+k ,n



(l)

(3−l)

|Ii,n ||Ij,n

l=1,2 i,j:t(l) ,t(3−l) ≤T

i+k,n

k=−Kn

≤n



Kn 

 

≤

j,n

−1 

(3−l)

|1{I (l) ∩I (3−l) =0}

i+k+m,n

(l)

|Ii,n ||Ij,n

i,n

j,n

j,n

0    (l) I

i+k,n

k=−Kn 2Gn 2,2 (T )

i,n

 (l) I

l=1,2 t(l) ,t(3−l) ≤T

×

|1{I (l) ∩I (3−l) =0}

m=−Kn



i,n

i+k,n )

j,n

Kn   (l) I

×

−1 (3−l) (l)  Mn (t

k =−Kn

k=−Kn

i,n

 (l) I

i+k

 

0 

 (l) I

i+m,n

Kn    (l) −1   I  i+m,n i+k,n

Kn −1   (l)  I +

m=−Kn

m=0

k=0

where we used the same index change trick as in (4.23).  P

 TDisJ as P → ∞ Step 3. Using dominated convergence, we obtain Y (P ) −→ D which allows to deduce       P    DisJ  Y (P ) ≤ ΥX 1 DisJ −→ P P DT ≤ ΥX 1ΩDisJ Ω T

T

(9.35)

for P → ∞. For details compare Step 3 in the proof of Proposition 4.10. Step 4. Following the arguments in Step 4 in the proof of Proposition 4.10 the claim (9.29) follows from (9.30), (9.31) and (9.35). Proof of Theorem 9.8. For proving (9.12) it is sufficient to show

 DisJ    nV (f, πn )T > V − (4, f(2,2) , πn , (β, ))T + Q  %DisJ P → α (9.36) T,n (1 − α) F with P(F DisJ ) > 0. To this end, note that we obtain for all F DisJ ⊂ ΩDisJ T  P

DisJ %DisJ Q (α)1ΩDisJ n,T (α)1ΩDisJ −→ QT T

T

(9.37)

as n → ∞ for each α ∈ [0, 1] from Proposition 9.12; compare Step 1 in the proof of Theorem 4.3.

9.2 Null Hypothesis of Joint Jumps

267

Then, Theorem 9.4 together with the convergences (9.10) and (9.37) yield using Lemma B.3 the X -stable convergence

%DisJ (nV (f, πn )T , V − (4, f(2,2) , πn , (β, ))T , Q n,T (1 − α)) L−s

TDisJ + D  TDisJ , C TDisJ , QDisJ −→ (C (1 − α)). T

from which we obtain using Lemma B.5 on ΩDisJ T

 DisJ   nV (f, πn )T > V − (4, f(2,2) , πn , (β, ))T + Q %DisJ P T,n (1 − α)} ∩ F  DisJ 

   D T →P > QDisJ (1 − α) ∩ F DisJ = αP F DisJ . T Here, the last equality follows from the definition of QDisJ (α). This convergence T implies (9.36) and hence (9.12). The consistency claim (9.13) follows from the fact that ΨDisJ n,T converges to a CoJ DisJ strictly positive limit on ΩT by Theorem 9.1 while cT,n (α) = oP (1). To see this note that we have DisJ cT,n (α) =

%DisJ n−1 V − (4, f(2,2) , πn , (β, ))T + n−1 Q T,n (1 − α) 

V (1) (g4 , πn )T V (2) (g4 , πn )T

where the denominator converges in probability to a strictly positive value and n−1 V − (4, f(2,2) , πn , (β, ))T vanishes as n → ∞ because big common jumps are eventually filtered out due to the indicator and because of (2.14). Further, it holds

%DisJ n−1 Q T,n (1 − α) ≤

Mn   DisJ  n−1 D % T,n,m  (1 − α)Mn  m=1

DisJ % T,n,m where E[|D ||S] = OP (1) follows using arguments from Step 2 in the proof of Proposition 9.12.

9.2 Null Hypothesis of Joint Jumps In this section we develop a statistical test which allows to decide whether common jumps are present or not in the paths of two stochastic processes under the null hypothesis that common jumps are present.

9.2.1 Theoretical Results Here, like in Section 9.1, we will work with functionals based on the function f(2,2) and we consider a ratio statistic very similar to the one used in Chapter 8 to test for the presence of jumps. In fact the methods presented in this section can be

268

9 Testing for the Presence of Common Jumps J,(l)

understood as a natural extension of the methods based on the statistic ΨT,n from Chapter 8 and we define ΨCoJ k,T,n =

V (f(2,2) , [k], πn )T k2 V (f(2,2) , πn )T

for natural numbers k ≥ 2. (l)

Remark 9.13. In the setting of equidistant observation times ti,n = i/n, l = 1, 2, the statistic ΨCoJ k,T,n is equal to

nT 

(1) i,j=k (Xi/n

(1)

(2)

(2)

− X(i−k)/n )2 (Xj/n − X(j−k)/n )2 1{((i−k)/n,n]∩((j−k)/n,j/n] =∅} k2

nT  i=1

(1)

(1)

(2)

(2)

(Xi/n − X(i−1)/n )2 (Xi/n − X(i−1)/n )2

.

In [32] a test for common jumps is constructed based on the statistic

nT /k i=1

(1)

(1)

(2)

(2)

(Xki/n − Xk(i−1)/n )2 (Xki/n − Xk(i−1)/n )2

nT  i=1

(1)

(1)

(2)

(2)

(Xi/n − X(i−1)/n )2 (Xi/n − X(i−1)/n )2

(9.38)

where at the lower observation frequency n/k only increments over the intervals (l) Iki,k,n , l = 1, 2, enter the estimation. Further in Section 14.1 of [3] a test for common jumps based on

nT 

(1) (1) (2) (2) 2 2 i=k (Xi/n − X(i−k)/n ) (Xi/n − X(i−k)/n ) nT  (1) (1) (2) (2) k i=1 (Xi/n − X(i−1)/n )2 (Xi/n − X(i−1)/n )2

(9.39)

is discussed. As argued in Remark 8.1 it seems advantegeous to use the statistic (9.39) over (9.38). However, in the asynchronous setting it is a priori not clear which observation intervals should be best paired, because there is no one-to-one correspondence of observation intervals in one process to observation intervals in the other process as there is in the synchronous situation. To use the available data as exhaustively as possible therefore here products of squared increments over all overlapping observation intervals at the lower observation frequency are included in the numerator of ΨCoJ  k,T,n . Under the null hypothesis that common jumps are present we obtain the following result from Theorem 2.3 and part b) of Theorem 2.40. Theorem 9.14. Suppose Condition 1.3 is fulfilled. Then it holds P

ΨCoJ k,T,n 1ΩCoJ −→ 1ΩCoJ . T

T

(9.40)

9.2 Null Hypothesis of Joint Jumps

269

Under the alternative ω ∈ ΩnCoJ the asymptotics of ΨCoJ T k,T,n are more complicated. In fact by Theorems 2.3 and 2.40 both numerator and denominator of ΨCoJ k,T,n vanish on ΩnCoJ . As in (8.3) we therefore expand the fraction by n to derive the T nCoJ asymptotics of ΨCoJ . As shown in the proof of Theorem 9.4 it holds k,T,n on ΩT L−s





 TDisJ 1 nCoJ TDisJ + D nV (f(2,2) , πn )T 1ΩnCoJ −→ C Ω T

T

and we will obtain a similar result for n/k2 V (f(2,2) , [k], πn )T . To describe the limit of n/k2 V (f(2,2) , [k], πn )T on ΩnCoJ and also the dependence structure of the T limits of n/k2 V (f(2,2) , [k], πn )T and nV (f(2,2) , πn )T we need to introduce some further notation. Recall the Brownian motion W t defined in (3.23) and set



(l)

ηk ,n,− (s) =

(3−l) j≥k :tj,n ≤T



(l)

ηk ,n,+ (s) =

(3−l)

j≥k :tj,n

DisJ,(l)

Zk ,n



1{s∈I (3−l) } j,k ,n

1{s∈I (3−l) } j,k ,n

≤T

(l)

(s) = ηk ,n,− (s), Δ t

(l) (l)

in (s),n

 

(l)

W

(l)

in (s)+1,n

DisJ,(l)

(l)

(Δi,k ,n W

n

(l) 2

) 1{i≥i(l) (s)+k } , n

(3−l)

(l)

for k = 1, k. Here it holds Z1,n

) 1{i 0. Further, 0

T

[k],n

[k],n

n (i) the functions Gn 2,2 (t), G2,2 (t), H0,0,4 (t), H0,0,4 (t), see (2.39) and (2.89) for their definition, converge pointwise on [0, T ] in probability to strictly [k] (t), H0,0,4 (t), H [k] (t) : [0, ∞) → [0, ∞). increasing functions G2,2 (t), G 2,2 0,0,4

(ii) The integral  [0,T ]P1 +P2

×

P2  p=1

g(x1 , . . . , xP1 , x 1 , . . . , x P2 )E

(2) 

hp

P1 

(1) 

hp

DisJ,(1)

nZ1,n

p=1

DisJ,(2)

nZ1,n

(x p ),



(xp ),

 n DisJ,(1) Z (xp ) k2 k,n

n DisJ,(2)  Z (xp ) dx1 . . . dxP1 dx 1 . . . dx P2 k2 k,n

270

9 Testing for the Presence of Common Jumps converges for n → ∞ to  [0,T ]P1 +P2

×

g(x1 , . . . , xP1 , x 1 , . . . , x P2 )

P2  

(1)

2 p=1 R

hp (y)ΓDisJ,[k],(1) (xp , dy)

(2)

hp (y)ΓDisJ,[k],(2) (x p , dy)dx1 . . . dxP1 dx 1 . . . dx P2

R2

p=1

P1  

(l)

for all bounded continuous functions g : RP1 +P2 → R, hp : R2(k+1) → R, p = 1, . . . , Pl , and any Pl ∈ N, l = 1, 2. Here ΓDisJ,[k],(l) (·, dy), l = 1, 2, are families of probability measures on [0, T ] where DisJ,(l)

(Z1

DisJ,(l)

(x), Zk

(x)) ∼ ΓDisJ,[k],(l) (x, dy)

has first moments which are uniformly bounded in x. Further the components DisJ,(l)

of Zk

(x) which correspond to the Δ

bounded second moments.

(l) (l)

in (x)+j,n

W

(l)

have uniformly

nCoJ In order to describe the limit of ΨCoJ we define k,T,n on ΩT

kDisJ C =  ,T



T 0

(1) (2)

 

 kDisJ D =  ,T

[k]

(σs σs )2 dG2,2 (s) +

p:Sp ≤T

(1)



T 0

(1) (2)

[k]

2(ρs σs σs )2 dH0,0,4 (s),

(2)

(2)



(1)

(ΔXSp )2 Rk (Sp ) + (ΔXSp )2 Rk (Sp ) ,

k = 1, k,

k = 1, k. (l)

Here, we denote by Sp , p ∈ N, an enumeration of the jump times of X and Rk (s) is defined via 

(l)

(l)

(l)

Rk (s) = (σs− )2 ηk ,− (s) +

k  

(l)

+ σs



(l)

(l)

δ+ (s)U+ (s) + σs

(l)

χj + σs−



j=−k +i

i=1 i−1 

−1 

(l)

σs−

2 χj

(l)

(l)

δ− (s)U− (s)

(l)

+ (σs )2 ηk ,+ (s), l = 1, 2, k = 1, k,

j=1

for random variables DisJ,(l)

(Z1

DisJ,(l)

(s), Zk

(l)



(l)

(l)

(s)) = (η1,− (s), δ− (s), δ+ (s), η1,+ (s)), (l)

(ηk,− (s), χ−k+1 , . . . , χ−1 , δ− (s), δ+ (s), χ1 , . . . , χk−1 , ηk,+ (s))



9.2 Null Hypothesis of Joint Jumps

271

) and whose distribution  F, P which are defined on an extended probability space (Ω, is given by DisJ,(l)

(Z1 P

DisJ,(l)

(s),Zk

(s))

(dy) = ΓDisJ,[k],(l) (s, dy), l = 1, 2.

DisJ,(l)

DisJ,(l)

The random variables (Z1 (s), Zk (s)), s ∈ [0, T ], are independent of each other and independent of the process X and its components. The random (l) (l) ) as well.  F, P variables U− (s), U+ (s) are i.i.d. standard normal and defined on (Ω, DisJ,(l)

DisJ,(l)

Furthermore, they are independent of F and of the (Z1 (s), Zk (s)). DisJ DisJ 1,T  1,T TDisJ and D  TDisJ occuring in Here, C and D correspond to the terms C Theorem 9.4. Theorem 9.16. Under Condition 9.15 we have the X -stable convergence L−s

ΨCoJ k,T,n −→

DisJ DisJ k,T  k,T +D kC

(9.41)

DisJ + D  DisJ C 1,T 1,T

. on ΩnCoJ T For the notion what it means for a sequence of random variables to converge X -stably in law on a subset of Ω see the paragraph following Theorem 8.8. From Theorem 9.14 we conclude that ΨCoJ k,T,n converges to 1 under the null hypothesis ω ∈ ΩCoJ and from Theorem 9.16 we obtain that ΨCoJ T k,T,n converges under the alternative ω ∈ ΩnCoJ to a limit which is almost surely different from 1 if T DisJ DisJ k,T 1,T kC = C , or under mild additional conditions if there is at least one jump

DisJ DisJ k,T 1,T in one of the components of X on [0, T ]. Indeed if we have kC = C , ΨCoJ k,T,n DisJ  DisJ DisJ k,T  k,T has to be different from 1 because C , C1,T are F -measurable and D

DisJ  1,T and D are either zero or their F -conditional distributions admit densities. If

DisJ DisJ DisJ DisJ k,T 1,T  k,T  1,T kC =C holds, mild conditions guarantee D = D almost surely

which also yields ΨCoJ k,T,n  1. Hence, similarly as in (8.6), we will construct a test with critical region



CoJ CoJ = |ΨCoJ Ck,T,n k,T,n − 1| > ck,T,n



(9.42)

CoJ for a (possibly random) sequence (ck,T,n )n∈N .

Like in the previous chapters we consider the situation where the observation times are generated by Poisson processes as an example for a random and irregular sampling scheme which fulfils the conditions we need for the testing procedure to work.

272

9 Testing for the Presence of Common Jumps

Example 9.17. Condition 9.15 is fulfilled in the setting of Poisson sampling; compare Definition 5.1. Indeed, part (i) of Condition 9.15 is fulfilled because the [k],n [k],n n functions Gn 2,2 (t), G2,2 (t), H0,0,4 (t), H0,0,4 (t) converge by Corollaries 5.6 and 5.8 to deterministic strictly increasing and linear functions. That part (ii) holds follows from Lemma 5.10.  CoJ Next, we derive a central limit theorem for ΨCoJ . However, before k,T,n on ΩT we start this endeavour, we restrict ourselves to the case k = 2. Already in this case, the form of the central limit theorem of ΨCoJ k,T,n is quite complicated. In the general case even the description of the limit would be very tedious due to the fact that we would have to keep track of numerous overlapping and non-overlapping interval parts. Further, from a practitioners perspective considering only k = 2 is not really a limitation as the simulation results in Section 8.2 for the test from Chapter 8 indicate that it is optimal to choose k as small as possible anyway. To state the upcoming central limit theorem for ΨCoJ k,T,n we need to introduce some notation. To this end we denote by CoJ,(l)

Ln

(s) = n|I

(s) = n|I LCoJ n

(l)

CoJ,(l)

(l)

in (s)−1,n

(1) (1)

in (s)−1,n

|, Rn

∩I

(2) (2)

in (s)−1,n

(s) = n|I

(l) (l)

in (s)+1,n

|, RCoJ (s) = n|I n

|,

l = 1, 2,

(1) (1)

in (s)+1,n

∩I

(2) (2)

in (s)+1,n

|

lengths of certain intervals around some time s at which a common jump might occur; for an illustration see Figure 9.5. Further we will use the shorthand notation



CoJ,(1)

ZnCoJ (s) = Ln

CoJ,(1)

, Rn

CoJ,(2)

, Ln

CoJ,(2)

, Rn



, LCoJ , RCoJ (s). n n

Note that, although they are of similar structure, the random variable ZnCoJ (s) differs both from Znbiv (s) and ZnCOV (s) defined in (3.20) repsectively (7.8) and used in Section 3.1 respectively Section 7.2. n−1 Ln

CoJ,(1)

(s)

 (1) I (1)

in (s),n

 

n−1 Rn

CoJ,(1)

(s)

X (1) s

n−1 LCoJ (s) n

n−1 RCoJ (s) n

X (2) n−1 Ln

CoJ,(2)

(s)

 (2) I (2)

in (s),n

 

n−1 Rn

Figure 9.5: Illustration of the components of ZnCoJ (s).

CoJ,(2)

(s)

9.2 Null Hypothesis of Joint Jumps

273

CoJ,(l)

CoJ,(l)

Limits of the variables Ln (s), Rn (s), l = 1, 2, and LCoJ (s), RCoJ (s) n n DisJ will occur in the central limit theorem for Ψ2,T,n . To ensure convergence of the ZnCoJ (s) we need to impose the following assumption on the observation scheme. [2],n

Condition 9.18. Assume that Condition 1.3 is fulfilled, that G2,2 = OP (1) holds √ and that we have n|πn |T = OP (1). Further suppose that the integral

 g(x1 , . . . , xP )E

[0,T ]P

P 



hp (ZnCoJ (xp )) dx1 . . . dxP

p=1

converges for n → ∞ to

 [0,T ]P

g(x1 , . . . , xP )

P   p=1 R

hp (y)ΓCoJ (xp , dy)dx1 . . . dxP

for all bounded continuous functions g : RP → R, hp : R6 → R and any P ∈ N. Here ΓCoJ (·, dy) is a family of probability measures on [0, T ] with uniformly bounded first moments. Using the limit distribution of ZnCoJ (s) implicitly defined in Condition 9.18 we set



ΦCoJ 2,T = 4 ×



p:Sp ≤T (2) ΔXSp

$



(1)

(1)

(2)

ΔXSp ΔXSp

σ Sp −

$

(1),−

LCoJ (Sp )USp

(1)

$

+ σ Sp

(1)

(1),+

RCoJ (Sp )USp

(1)

(2)

(σSp − )2 (LCoJ,(1) − LCoJ )(Sp ) + (σSp )2 (RCoJ,(1) − RCoJ )(Sp )USp

+

(1)



(2)

$

$

+ +

(1),−

LCoJ (Sp )USp

+ ΔXSp σSp − ρSp −

(2)

$

+ σ Sp ρSp

(2)



(1),+

RCoJ (Sp )USp

(2)

(3)

(σSp − )2 (1 − (ρSp − )2 )LCoJ (Sp ) + (σSp )2 (1 − (ρSp )2 )RCoJ (Sp )USp

$

(2)

(2)

(4)

(σSp − )2 (LCoJ,(2) − LCoJ )(Sp ) + (σSp )2 (RCoJ,(2) − RCoJ )(Sp )USp

 .

Here, we denote by Sp , p ∈ N, an enumeration of the common jump times of X (1) and X (2) , the vector





LCoJ,(1) , RCoJ,(1) , LCoJ,(2) , RCoJ(2) , LCoJ , RCoJ (s)

is distributed according to ΓCoJ (s, ·) and the vectors



(1),−

Us

(1),+

, Us

(2)

(3)

(4) 

, U s , U s , Us

274

9 Testing for the Presence of Common Jumps

are standard normally distributed and independent for different values of s. Similarly as for (8.8) we obtain that the infinite sum in the definition of ΦCoJ 2,T has a well-defined limit. Using the variable ΦCoJ we are able to state the following 2,T central limit theorem. Theorem 9.19. If Condition 9.18 is fulfilled, we have the X -stable convergence

 √  CoJ L−s n Ψ2,T,n − 1 −→

ΦCoJ 2,T ∗ 4B (f(2,2) )T

(9.43)

. on the set ΩCoJ T For the notion what it means for a sequence of random variables to converge X -stably in law on a subset of Ω see the paragraph following Theorem 8.8. Example 9.20. Condition 9.18 is fulfilled in the Poisson setting. The property √ [2],n n|πn |T = OP (1) follows from (5.2) and G2,2 = OP (1) holds true due to Corollary 5.8 as discussed in Example 9.17. Further the convergence of the integrals follows like part (ii) of Condition 9.15 from Lemma 5.10.  Example 9.21. Condition 9.18 is also fulfilled in the setting of equidistant obser(1) (2) vation times ti,n = ti,n = i/n. In that case we have





LCoJ,(1) , RCoJ,(1) , LCoJ,(2) , RCoJ,(2) , LCoJ , RCoJ (s) = (1, 1, 1, 1, 1, 1)

for any s ∈ (0, T ]. Hence we get



ΦCoJ 2,T = 4

(1)

p:Sp ≤T

(2)



(2)  (1)

(1),−

ΔXSp ΔXSp ΔXSp σSp − USp (1)  (2)

(1),−

+ ΔXSp σSp − ρSp − USp

$

+

(2)

(1)

(1),+ 

+ σ S p US p (1),+

+ σ S p ρ S p US p

(2)

(3) 

(2)

(σSp − )2 (1 − (ρSp − )2 ) + (σSp )2 (1 − (ρSp )2 )USp



and ΦCoJ 2,T is X -conditionally normally distributed with mean zero and variance 16

 p:Sp ≤T

(1)

(2)

(ΔXSp )2 (ΔXSp )2



(2) (1)



(2) (1)

(1) (2)

+ ΔXSp σSp + ΔXSp σSp ρSp (2)

(1) (2)

ΔXSp σSp − + ΔXSp σSp − ρSp −

+ (σSp )2 (1 − (ρSp )2 )



2

(1)



2

(2)

+ (ΔXSp )2 (σSp − )2 (1 − (ρSp − )2 )

which is similar to the result in Theorem 4.1(a) of [32].



9.2 Null Hypothesis of Joint Jumps

275

Using the central limit theorem 9.19 and the methods from Chapter 4 we are  n )n∈N and (Mn )n∈N finally able to construct the statistical test. To this end let (K be sequences of natural numbers which tend to infinity and set



(l)

%n,m L

(s) = nI

% n,m R

(s) = nI

CoJ,(l)

CoJ,(l)



 (l)  %CoJ L n,m (s) = n I (1) 

(l)

(l)

in (s)+Vn,m (s)−1,n (l) (l)

(l)

in (s)+Vn,m (s)+1,n

(1) in (s)+Vn,m (s)−1,n

 % CoJ R n,m (s) = n I

(l) (1) (1) in (s)+Vn,m (s)+1,n

(1)

 , l = 1, 2,  , l = 1, 2,

∩I

(l) (2) (2) in (s)+Vn,m (s)−1,n

∩I

 

(l) (2) (2) in (s)+Vn,m (s)+1,n

(9.44)

 

(2)

n, . . . , K  n } × {−K n, . . . , K  n } and where Vn,m (s) = (Vn,m (s), Vn,m (s)) ∈ {−K     P Vn,m (s) = k˜1 , k˜2 S = |I

(1) (1) ˜1 ,n in (s)+k

∩I

(2) (2) ˜2 ,n in (s)+k

|





Kn 

|I

 ˜  ,k ˜ k 1 2 =−Kn

(1) (1) ˜  ,n in (s)+k 1

∩I

(2) (2) ˜  ,n in (s)+k 2

|

−1

(9.45) ˜1 , k ˜2 ∈ {−K n, . . . , K  n }. Based on (9.44) we define via for k



CoJ %n,m %n,m Z (s) = L

CoJ,(1)

% n,m ,R

CoJ,(1)

%n,m ,L

CoJ,(2)

% n,m ,R

CoJ,(2)



%CoJ % CoJ ,L n,m , Rn,m (s),

m = 1, . . . , Mn , estimators for realizations of Z CoJ (s). Using the components of CoJ %n,m Z (s) and considering increments which are larger than a certain threshold as jumps, compare Section 2.3, we then define



% CoJ Φ 2,T,n,m = 4

(1)

(1)

(2)

Δi,n X (1) Δj,n X (2)

(2)

i,j:ti,n ∨tj,n ≤T

× 1{|Δ(1) X (1) |>β|I (1) | } 1{|Δ(2) X (2) |>β|I (2) | } ×



i,n

(2) Δj,n X (2)

i,n



σ ˜

(1)

(1) (ti,n , −) (1)



$

j,n

j,n

(1,2),−

n %CoJ L n,m (τi,j )Un,(i,j),m

+σ ˜ (1) (ti,n , +)

$

(1)

(1,2),+

n % CoJ R n,m (τi,j )Un,(i,j),m

%n,m + (˜ σ (1) (ti,n , −))2 (L

CoJ,(1)

(1)

(1)



n %CoJ −L n,m )(τi,j )

% n,m + (˜ σ (1) (ti,n , +))2 (R (2)

n + Δi,n X (1) σ ˜ (2) (tj,n , −)˜ ρ(τi,j , −)

CoJ,(1)

$

n % CoJ −R n,m )(τi,j ) (1,2),−

n %CoJ L n,m (τi,j )Un,(i,j),m

1/2

(1)

Un,i,m



276

9 Testing for the Presence of Common Jumps (2)



n +σ ˜ (2) (tj,n , +)˜ ρ(τi,j , +)

$

(1,2),+

n % CoJ R n,m (τi,j )Un,(i,j),m

(2)

n n %CoJ + (˜ σ (2) (tj,n , −))2 (1 − (˜ ρ(τi,j , −))2 )L n,m (τi,j ) (2)



n n % CoJ + (˜ σ (2) (tj,n , +))2 (1 − (˜ ρ(τi,j , +))2 )R n,m (τi,j ) (2)

%n,m + (˜ σ (2) (tj,n , −))2 (L

CoJ,(2)

(2)

1/2

(2)

Un,j,m

n %CoJ −L n,m )(τi,j )

% n,m + (˜ σ (2) (tj,n , +))2 (R

CoJ,(2)

n % CoJ −R n,m )(τi,j )

1/2

(3)



Un,j,m

× 1{I (1) ∩I (2) =∅} i,n

(1)

j,n

(2)

n with τi,j = ti,n ∧ tj,n . Here, we use the estimators σ ˜ (l) (s, −), σ ˜ (l) (s, +), l = 1, 2, (l)

respectively ρ˜(s, −), ρ˜(s, +) introduced in Chapter 6 for the estimation of σs− ,

(l) σs ,

(1,2),− Un,(i,j),m ,

(1,2),+ Un,(i,j),m ,

(1) Un,i,m ,

(2) Un,j,m ,

(3) Un,j,m

l = 1, 2, and ρs− , ρs . The are i.i.d. standard normal distributed random variables and we choose β > 0 and  ∈ (0, 1/2). Further, we denote by

%CoJ % Q 2,T,n (α) = Qα





 % CoJ |Φ 2,T,n,m | m = 1, . . . , Mn



% CoJ the αMn -th largest element of the set {|Φ 2,T,n,m ||m = 1, . . . , Mn } and we

CoJ %CoJ under appropriate conditions to the will see that Q 2,T,n (α) converges on ΩT CoJ CoJ X -conditional α-quantile Q2,T (α) of |Φ2,T | which is defined via

   CoJ CoJ  |ΦCoJ  P , 2,T | ≤ Q2,T (α) X (ω) = α, ω ∈ ΩT 



nCoJ and we set QCoJ , for completeness. Such a random vari2,T (α) (ω) = 0, ω ∈ ΩT CoJ able Q2,T (α) exists because Condition 9.23 will guarantee that the X -conditional CoJ distribution of ΦCoJ . 2,T is almost surely continuous on ΩT

% CoJ Remark 9.22. Note that we could also define the estimator Φ 2,T,n,m as the sum (1)

(2)

over all terms where in the product Δi,n X (1) Δj,n X (2) at least one increment has to be large. This difference has asymptotically no effect because the contribution of idiosyncratic jumps vanishes in the limit. In the corresponding test based on equidistant and synchronous observations from [32] this alternative approach is taken. Although idiosyncratic jumps play no role asymptotically they contribute % CoJ to the approximation error. Therefore including them in the estimator Φ 2,T,n,m helps to diminish this approximation error because we estimate part of it. On the % CoJ contrary, this approach yields that the sum in Φ 2,T,n,m includes on average much more terms which slows down the numerical implementation of the test. With the simulation study, which will be presented in Section 9.2.2, in mind we therefore

9.2 Null Hypothesis of Joint Jumps

277

chose only to include terms where increments of both processes are identified as jumps.  The necessary assumptions to obtain a valid statistical test are summarized in the following condition. Condition 9.23. The process X and the sequence of observation schemes (πn )n∈N fulfil Condition 9.15 and 9.18. Further the set {s ∈ [0, T ] : σs  = 0} is almost T  n )n∈N and (Mn )n∈N surely a Lebesgue null set and 0 ΓCoJ (x, {0}6 )dx = 0. (K  n /n → 0 and the are sequences of natural numbers converging to infinity with K P

sequence (bn )n∈N fulfils |πn |T /bn −→ 0. Additionally, (i) for any xp ∈ R6 , sp ∈ (0, T ), p = 1, . . . , P , P ∈ N, with sp = sp for p = p

it holds P     CoJ   (Z CoJ (sp ) ≤ xp ) > ε → 0 P P(Z%n,1 P (sp ) ≤ xp , p = 1, . . . , P |S) − p=1

as n → ∞ for all ε > 0. n (ii) We have n1/2 (|πn |T )1− = oP (1) and Gn 1,2 (T ) = OP (1), G2,1 (T ) = OP (1).

(iii) The process σ is a 2 × 2-dimensional matrix-valued Itˆ o semimartingale. DisJ DisJ DisJ DisJ 2,T 1,T  2,T  1,T (iv) It holds 2C = C or D = D almost surely. CoJ %CoJ Part (i) of Condition 9.23 yields that Q 2,T,n (α) consistently estimates Q2,T (α). Part (ii) is a technical condition which will be needed in the proof of Theorem 9.24. (l) (l) Part (iii) provides that σ ˜n (Sp , −), σ ˜n (Sp , +), ρ˜n (Sp , −), ρ˜n (Sp , +) are consistent (l) (l) estimators of σSp − , σSp , ρSp − , ρSp , and part (iv) guarantees that the limit under the alternative is almost surely different from 1. Note also that Condition 9.23(iii) in general does not follow from the convergence of Gn 2,2 (T ) assumed in Condition 9.18.

Theorem 9.24. Let Condition 9.23 be fulfilled. Then the test defined in (9.42) with CoJ c2,T,n (α) = √

%CoJ Q 2,T,n (1 − α) n4V (f(2,2) , πn )T

, α ∈ [0, 1],

α ∈ [0, 1],

has asymptotic level α in the sense that we have

 CoJ   CoJ  |ΨCoJ  P → α α ∈ (0, 1], 2,T,n − 1| > c2,T,n (α) F

(9.46)

278

9 Testing for the Presence of Common Jumps

for all F CoJ ⊂ ΩCoJ with P(F CoJ ) > 0. Further the test is consistent in the T sense that we have

 nCoJ   CoJ  |ΨCoJ  P →1 2,T,n − 1| > c2,T,n (α) F

(9.47)

with P(F nCoJ ) > 0. for all F nCoJ ⊂ ΩnCoJ T To improve the performance of our test in the finite sample we adjust the estimator ΨCoJ 2,T,n , similarly as was done in Chapter 8 to obtain the test in Corollary 8.12, by (partially) subtracting terms that contribute in the finite sample but vanish asymptotically. Therefore we define for β,  as above the quantity ACoJ 2,T,n = n

 (1)

(1)

(2)

(2)

(Δi,2,n X (1) )2 (Δj,2,n X (2) )2 1{I (1)

(2) i,2,n ∩Ij,2,n =∅}

i,j:ti,n ∨tj,n ≤T



− 4n

× 1{|Δ(1)

i,2,n X

(1) |≤β|I (1) | ∨|Δ(2) X (2) |≤β|I (2) | } i,2,n j,2,n j,2,n

(1)

(1) (2) i,j:ti,n ∨tj,n ≤T

(2)

(Δi,n X (1) )2 (Δj,n X (2) )2 1{I (1) ∩I (2) =∅} i,n

j,n

× 1{|Δ(1) X (1) |≤β|I (1) | ∨|Δ(2) X (2) |≤β|I (2) | } . i,n

i,n

j,n

j,n

Using this expression we then define for ρ ∈ (0, 1) by CoJ  CoJ Ψ 2,T,n (ρ) = Ψ2,T,n − ρ

n−1 ACoJ 2,T,n 4V (f(2,2) , πn )T

the adjusted estimator. Corollary 9.25. Let ρ ∈ (0, 1). If Condition 9.23 is fulfilled, then it holds with the notation of Theorem 9.24

  CoJ  CoJ  |Ψ  2,T,n (ρ) − 1| > c2,T,n P (α)F CoJ → α α ∈ [0, 1],

(9.48)

for all F CoJ ⊂ ΩCoJ with P(F CoJ ) > 0 and T

  CoJ  CoJ  |Ψ  2,T,n (ρ) − 1| > c2,T,n P (α)F nCoJ → 1 α ∈ (0, 1],

(9.49)

with P(F nCoJ ) > 0. for all F nCoJ ⊂ ΩDisJ T Example 9.26. The assumptions on the observation scheme from Condition 9.23 are fulfilled in the setting of Poisson sampling discussed in Examples 9.17 and 9.20. That part (i) of Condition 9.23 holds follows from Lemma 5.12. Part (ii) is fulfilled by (5.2) (note that 1 −  > 1/2 due to  ∈ (0, 1/2)) and Corollary 5.6. T The property 0 ΓCoJ (x, {0}6 )dx = 0 follows from the considerations in Remark 5.11 

9.2 Null Hypothesis of Joint Jumps

279

Similarly as discussed in Chapter 8 after Example 8.13 we can omit the weighting in (9.45) for obtaining a working testing procedure also within the bivariate Poisson setting. However, in the setting where both processes X (1) and X (2) are observed at the observation times introduced in Example 8.14 with different α1 = α2 it can be easily verified that the weighting in (9.45) is necessary and leads to a correct estimation of the distribution of ΦCoJ 2,T .

9.2.2 Simulation Results To evaluate the finite sample performance of the tests described in Theorem 9.24 and in Corollary 9.25 we conduct a simulation study. Here, we use the same model and parameter settings as in Section 9.1.2; compare Table 9.1.

1.0

1.0

1.0 0.0

0.0

0.5

1.0

III-d0

1.0

1.0

0.0

0.0 0.5

0.0

0.5

1.0

0.5

1.0

II-d1

0.0

0.5

1.0

III-d1

1.0

0.5

III-m

0.0

0.0

0.5 0.0 0.0

0.5

1.0 0.5 0.0 0.0

1.0

0.5

1.0 1.0

0.5

II-d0

0.5

0.5

III-j

0.0

1.0

1.0

0.0

0.5 0.0 0.0

0.5

1.0 0.5

0.5

II-m

0.5

1.0

0.0

0.5

1.0

I-d1

0.0

0.5

II-j

1.0

0.0

I-d0

0.0

0.5

1.0

I-m

0.0

0.0

0.5

1.0

I-j

0.0

0.5

1.0

0.0

0.5

Figure 9.6: Empirical rejection curves from the Monte Carlo simulation for the test derived from Theorem 9.24. The dotted lines represent the results for n = 100, the dashed lines for n = 1,600 and the solid lines for n = 25,600. In each case N = 10,000 paths were simulated.

1.0

280

9 Testing for the Presence of Common Jumps

I-m

I-d0

I-d1

3

4

3.0 2.0 1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.5

1.5

II-m

2.5

3.5

2

3.0 2.0

8

4

2

6

8

3

II-d1

4

1.0

6 0.5

1.0

1.5

2.0

2.5

3.0

0.5

1.0

1.5

2.0

2.5

3.0

0.5

III-m

1.5

2.5

3.5

0.5

1.5

III-d0

2.5

3.5

III-d1

0.5

1.0

1.5

2.0

2.5

3.0

4 1

0.5

1.0

1.5

2.0

2.5

3.0

0

0

0

0

2

2

1

2

4

4

2

3

6

6

3

8

8

5

4

10

6

5

7

12

10 12 14

III-j

0

0.0

0

0

2

2

1

4

1

II-d0

10 12

10 12 14

II-j

4

1.0

3

0.5

0

0.0

0

0

1

5

5

2

10

10

15

15

5

I-j

0.5

1.5

2.5

3.5

0.5

1.5

2.5

3.5

Figure 9.7: Density estimates for ΦCoJ 2,T,n from the Monte Carlo. The dotted lines correspond to n = 100, the dashed lines to n = 1,600 and the solid lines to n = 25,600. In all cases N = 10,000 paths were simulated. In Figures 9.6 and 9.7 we display the results from the simulation study for the testing procedure from Theorem 9.24 and in Figures 9.8 and 9.9 for the test from Corollary 9.25. First we plot in Figure 9.6 for all twelve cases the empirical rejection rates from Theorem 9.24 in dependence of α ∈ [0, 1] as for the plots in Figure 9.3. The six plots on the left show the results for the cases where the hypothesis of the presence of common jumps is true. Similarly as in Sections 8.2 and 9.1.2 we observe that the empirical rejection rates match the postulated asymptotic level of the test better if n is larger or if common jumps are larger on average. In all six cases where common jumps are present and for all values of n the test overrejects. This is due to the fact that the asymptotically negligible CoJ terms in ΨCoJ 2,T,n (terms which are contained in Ψ2,T,n but do not contribute in the limit) tend to be positive, hence ΨCoJ 2,T,n is on average systematically larger than 1, see Figure 9.7, which yields the bias. However, at least for n = 25,600

9.2 Null Hypothesis of Joint Jumps

281

the observed rejection rates match the asymptotic level quite well. Only in the cases III-j and III-m, where the jumps are on average very small, the empirical rejection rates are still far higher than the asymptotic level. The results are worse in the mixed model than in the model where there are only common jumps because idiosyncratic jumps contribute to the asymptotically vanishing error. The test has very good power against the alternative of idiosyncratic jumps as can be seen in the six plots on the right-hand side of Figure 9.6 which correspond to the cases where there are no common jumps. Figure 9.7 shows density estimates for ΨCoJ 2,T,n in all twelve cases. If there are common jumps it is visible in the density plots that ΨCoJ 2,T,n converges to 1 as n → ∞. However for n = 100, n = 1,600 in the cases II-j, II-m and for n = 100, n = 1,600, n = 25,600 in the cases III-j, III-m the density peaks at a value significantly larger than 1 which corresponds to the overrejection in Figure 9.6. Under the alternative of disjoint jumps ΨCoJ 2,T,n tends to cluster around 1.5 which corresponds to the results obtained in Example 8.5 in the univariate setting. Our simulation results from Theorem 9.24 are worse than the results in the equidistant setting displayed in Figure 4 of [32] while the power of our test is much better. This effect is partly due to the fact that, contrary to our approach, in [32] idiosyncratic jumps, although their contribution is asymptotically negligible, are included in the estimation of the asymptotic variance in the central limit theorem; compare Remark 9.22. Hence they consistently overestimate the asymptotic variance which yields lower rejection rates. Further, the asymptotically negligible terms in ΨCoJ 2,T,n are larger relative to the asymptotically relevant terms in the asynchronous setting than in the setting of synchronous observation times which also increases the rejection rates in the asynchronous setting. The test from Corollary 9.25 outperforms the test from Theorem 9.24 in the simulation study. This can be seen in Figures 9.8 and 9.9 which show the results from the Monte Carlo simulation for the test from Corollary 9.25 with ρ = 0.75 (for the choice of ρ = 0.75 see Figure 9.10) in the same fashion as for Theorem 9.24 in Figures 9.6 and 9.7. In the cases where common jumps are present we observe that the empirical rejection rates match the asymptotic level much better than in Figure 9.6. In Figure 9.8 we see that in the cases I-j, I-m, II-j, II-m we get good results already for n = 1,600 and in the cases III-j, III-m at least for n = 25,600. The power of the test from Corollary 9.25 is practically as good as for  CoJ the test from Theorem 9.24. Hence using the adjusted statistic Ψ 2,T,n (ρ) instead CoJ of Ψ2,T,n allows to get far better level results while the power of the test remains almost the same.  CoJ Figure 9.9 shows that the adjusted estimator Ψ 2,T,n (ρ) is much more centered CoJ around 1 than Ψ2,T,n if there exist common jumps which is most pronounced in

 CoJ the cases III-j and III-m. On the other hand, Ψ 2,T,n (ρ) clusters around a value very close to 1 also if there exist no common jumps. However, in all cases *-d0

9 Testing for the Presence of Common Jumps

1.0

0.0 0.5

1.0

III-d0

1.0

1.0

0.0

0.0 0.5

0.0

0.0

0.5

1.0

0.0

1.0

0.5

1.0

III-d1

1.0

1.0

0.5

II-d1

0.5

0.5

III-m

0.0

0.0

0.0 0.0

0.5

1.0 0.5 0.0 0.0

1.0

0.5

1.0 1.0

0.5

II-d0

0.5

0.5

III-j

0.0

1.0

1.0

0.5 0.0

1.0

1.0 0.5

0.5

II-m

0.0

0.0

0.5

1.0

0.0

I-d1

0.0

1.0

1.0

0.5

II-j

0.5

0.0

I-d0

0.0

0.5

1.0

I-m

0.0

0.0

0.5

1.0

I-j

0.5

282

0.0

0.5

1.0

0.0

0.5

1.0

Figure 9.8: Empirical rejection curves from the Monte Carlo simulation for the test derived from Corollary 9.25. The dotted lines represent the results for n = 100, the dashed lines for n = 1,600 and the solid lines for n = 25,600. In each case N = 10,000 paths were simulated. and *-d1 the peak of the density still occurs at a value which is noticeably larger than 1. Figure 9.10 illustrates similarly as Figure 8.3 for the test for jumps the performance of the test from Corollary 9.25 in dependence of ρ. We choose the cases III-j and III-d0 as representatives for the null hypothesis and the alternative. As expected the level as well as the power of the test decrease as ρ increases. Here we get the lowest overall error for a value of ρ close to 0.75. As demonstrated in Section 8.2 the adjustment method leading to the test in Corollary 9.25 can also be used to improve the finite sample performance of existing tests based on equidistant and synchronous observations. To justify this claim we repeated the above simulation study for observation times given by (1) (2) ti,n = ti,n = i/n. The results are presented in Figures 9.11–9.15 in the same

9.2 Null Hypothesis of Joint Jumps

I-m

I-d0

I-d1

30

40

15

1.0

1.5

2.0

2.5

3.0

0.5

1.0

1.5

2.0

2.5

3.0

0 0.5

1.5

II-m

2.5

3.5

2.0

2.5

40 20 10

10 1.0

1.5

2.0

2.5

3.0

0

5 0 0.5

0.5

1.5

III-m

2.5

3.5

0.5

1.5

III-d0

2.5

3.5

III-d1

0.5

1.0

1.5

2.0

2.5

0.5

1.0

1.5

2.0

0

0

0

0

2

2

5

5

4

4

6

6

10

8

10

8

15

15

10 12

20

III-j 10 12 14

4

30

20 1.5

3

15

8 6 4 2 0 1.0

2

II-d1

25

10 12

10 12 14 8 6 4 2 0 0.5

1

II-d0 30

II-j

50

0.5

0

0

0

10

5

5

5

20

10

10

10

15

15

50

I-j

283

0.5

1.0

1.5

2.0

2.5

0.5

1.0

1.5

2.0

2.5

 CoJ Figure 9.9: Density estimates for Ψ 2,T,n (ρ) from the Monte Carlo. The dotted lines correspond to n = 100, the dashed lines to n = 1,600 and the solid lines to n = 25,600. In all cases N = 10,000 paths were simulated.

fashion as above. Like in the setting of Poisson sampling we observe that the level of the test is much closer to the ideal asymptotic value for the test based on the CoJ  CoJ adjusted statistic Ψ 2,T,n (ρ) compared to the test based on Ψ2,T,n . Figure 9.15 shows that, for the here chosen parameter settings, the optimal value ρ ≈ 0.79 under equidistant and synchronous observations is very close to the optimal value ρ ≈ 0.75 in the Poisson setting.

9 Testing for the Presence of Common Jumps

1.0

284

0.2

0.4

0.6

0.8

Level 1-Power (Level+(1-Power))

0.0

0.05

0.0

0.2

0.4

0.6

0.8

1.0

ρ

Figure 9.10: This graphic shows for α = 5% and n = 25,600 the empirical rejection rate in the case III-j and 1 minus the empirical rejection rate in the case III-d0 from the Monte Carlo simulation based on Corollary 9.25 as a function of ρ ∈ [0, 1]. We achieve a minimal overall error for ρ ≈ 0.75.

1.0

1.0 0.0

0.5

1.0

III-d0

1.0

1.0

0.0

0.0 0.5

0.0

0.0

0.5

1.0

0.0

1.0

0.5

1.0

III-d1

1.0

1.0

0.5

II-d1

0.5

0.5

III-m

0.0

0.0

0.0 0.0

0.5

1.0 0.5 0.0 0.0

1.0

0.5

1.0 1.0

0.5

II-d0

0.5

0.5

III-j

0.0

1.0

1.0

0.5 0.0

0.5

1.0 0.5

0.5

II-m

0.0

0.0

0.5

1.0

0.0

I-d1

0.0

1.0

1.0

0.5

II-j

0.5

0.0

I-d0

0.0

0.5

1.0

I-m

0.0

0.0

0.5

1.0

I-j

0.0

0.5

1.0

0.0

0.5

1.0

Figure 9.11: Empirical rejection curves as in Figure 9.6 from the Monte Carlo simulation for the test derived from Theorem 9.24 based on (l) equidistant observations ti,n = i/n.

9.2 Null Hypothesis of Joint Jumps I-d0

2.5

3 2 1 3

4

3.0

0.5

1.0

1.5

2.0

1.5

2.5

2.5

3.5

II-d1 5 4 3 2 1

3.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0

III-m

0.5

III-d0

1.5

2.5

3.5

III-d1

3 1.0

6 1.0

1.5

2.0

2.5

3.0

0.5

1.0

1.5

2.0

2.5

3.0

1 0

0.0

0

0

2

2

0.5

4

4

2

6

8

4

1.5

8

10 12

10

III-j

0.5

0

6 4 2 0 2.0

2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

10 8

10 12 8 6

1.5

1

II-d0

4 2

1.0

0

II-m

0 0.5

0

5 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

II-j

4

0.0 0.2 0.4 0.6 0.8 1.0

15 10

10 5 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

I-d1 5

I-m

15

I-j

285

1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Figure 9.12: Density estimates for ΨCoJ 2,T,n as in Figure 9.7 from the Monte (l) Carlo based on equidistant observations ti,n = i/n.

1.0

1.0 0.0

0.5

1.0

III-d0

1.0

1.0

0.0

0.0 0.5

0.0

0.0

0.5

1.0

0.0

1.0

0.5

1.0

III-d1

1.0

1.0

0.5

II-d1

0.5

0.5

III-m

0.0

0.0

0.0 0.0

0.5

1.0 0.5 0.0 0.0

1.0

0.5

1.0 1.0

0.5

II-d0

0.5

0.5

III-j

0.0

1.0

1.0

0.5 0.0

0.5

1.0 0.5

0.5

II-m

0.0

0.0

0.5

1.0

0.0

I-d1

0.0

1.0

1.0

0.5

II-j

0.5

0.0

I-d0

0.0

0.5

1.0

I-m

0.0

0.0

0.5

1.0

I-j

0.0

0.5

1.0

0.0

0.5

1.0

Figure 9.13: Empirical rejection curves as in Figure 9.8 from the Monte Carlo simulation for the test derived from Corollary 9.25 for ρ = 0.75 (l) based on equidistant observations ti,n = i/n.

286

9 Testing for the Presence of Common Jumps I-m

I-d0

I-d1

0.0 0.5 1.0 1.5 2.0 2.5 3.0

15

6

10

5

5

2 0.0 0.5 1.0 1.5 2.0 2.5 3.0

0

1

2

3

4

0.5

1.5

II-d0

2.5

3.5

II-d1

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

30 20 0 0.5

1.5

III-m

2.5

3.5

1.5

6 5 4

8

2.5

3.5

III-d1

0.5

1.0

1.5

2.0

2.5

3.0

4 0

2

1 0

0

0

2

2

4

2

4

6

6

3

6

8

0.5

III-d0

10

10 12 14

III-j

8 10 12 14

0.5

0

0

0

2

2

2

10

4

4

4

6

6

6

8

8

8

10

10

II-m

10 12

II-j

0

1 0

0

0

2

5

4

3

6

4

8

10

15

10 12

7

I-j

0.5

1.0

1.5

2.0

2.5

3.0

0.5

1.5

2.5

3.5

0.5

1.5

2.5

3.5

1.0

 CoJ Figure 9.14: Density estimates as in Figure 9.9 for Ψ 2,T,n (0.75) from the (l) Monte Carlo based on equidistant observations ti,n = i/n.

0.2

0.4

0.6

0.8

Level 1-Power (Level+(1-Power))

0.0

0.05

0.0

0.2

0.4

0.6

0.8

1.0

ρ

Figure 9.15: This graphic shows for α = 5%, n = 25,600 and equidistant observations ti,n = i/n the empirical rejection rate in the case III-j along with 1 minus the empirical rejection rate in the case III-d0 from the Monte Carlo simulation based on Corollary 9.25 as a function of ρ ∈ [0, 1]. We achieve a minimal overall error for ρ ≈ 0.79.

9.2 Null Hypothesis of Joint Jumps

287

9.2.3 The Proofs Proof of Theorem 9.14. By Theorems 2.3 and by Theorem 2.40 b) we obtain P

P

V (f(2,2) , πn )T −→ B ∗ (f(2,2) )T and V (f(2,2) , [k], πn )T −→ k2 B ∗ (f(2,2) )T . These two convergences and the continuous mapping theorem for convergence in probability yield (9.40). Proof of Theorem 9.16. It holds ΨCoJ k,T,n =

n/k2 V (f(2,2) , [k], πn )T nV (f(2,2) , πn )T

and the proof of (n/k2 V (f(2,2) , [k], πn )T , nV (f(2,2) , πn )T ) L−s

DisJ DisJ  DisJ DisJ k,T  k,T  1,T −→ (kC +D , C1,T + D )

(9.50)

restricted to ΩnCoJ is similar to the proof of Theorem 9.4. For example we obtain T





n/k2

(1)

(1)

(2)

(2)

(Δi,k,n C (1) )2 (Δj,k,n C (2) )2 1{I (1)

(2) i,k,n ∩Ij,k,n =∅}

i,j:ti,n ∨tj,n ≤T

n

 (1)

(1)

(2)



(2)

,

P

DisJ  DisJ k,T (Δi,n C (1) )2 (Δj,n C (2) )2 1{I (1) ∩I (2) =∅} −→ (kC , C1,T )

i,j:ti,n ∨tj,n ≤T

i,n

j,n

from Theorem 2.22 and Theorem 2.42 b). Using Lemma B.5 we obtain (9.41) from T (1) (2) (9.50) because by Condition 9.15 we have 0 |σs σs |ds > 0 which together with DisJ DisJ DisJ 1,T  1,T 1,T the fact that G2,2 is strictly increasing guarantees C +D ≥C > 0. Proof of Theorem 9.19. We prove

 L−s √  n V (f(2,2) , [2], πn )T − 4V (f(2,2) , πn )T −→ ΦCoJ 2,T

(9.51)

from which (9.43) easily follows.  q) of Step 1. Recall the discretized versions σ ˜ (1) (r, q), σ ˜ (2) (r, q), ρ˜(r, q) and C(r, (1) (2) σ , σ , ρ and C defined in Step 1 in the proof of Theorem 3.6. We then show  √   q, r)| > ε) = 0 lim lim sup lim sup P(| n V (f(2,2) , [2], πn )T − 4V (f(2,2) , πn )T − R(n,

q→∞ r→∞

n→∞

(9.52)

288

9 Testing for the Presence of Common Jumps

for all ε > 0 where

 q, r) = R (1) (n, q, r) + R (2) (n, q, r), R(n,   (l) √ (l)  (l) (l) (n, q, r) = n 2 Δi−1,1,n N (l) (q)Δi,1,n C (r, q) R (l)

(3−l)

i,j≥2:ti,n ∧tj,n

≤T

  (l) (l) (r, q)Δ(l) N (l) (q) Δ(3−l) N (3−l) (q) 2 1 (l) , + Δi−1,1,n C (3−l) i,1,n j,2,n {Ii,2,n ∩Ij,2,n =∅} l = 1, 2. As the proof of (9.52) is rather technical it is postponed to after the discussion of the structure of the proof of (9.51). Step 2. Next, we will show L−s

 q, r) −→ ΦCoJ R(n, (q, r) T

(9.53)

for any r, q > 0 where ΦCoJ (q, r) := 4 T



(2)



(1)

p:Sq,p ≤T



× ΔXSq,p σ ˜ (1) (r, q)Sq,p − + +˜ σ (1) (r, q)Sq,p

(2)

ΔXSq,p ΔXSq,p

$

$

(1),−

LCoJ (Sq,p )USq,p (1),+

RCoJ (Sq,p )USq,p

+ (˜ σ (1) (r, q)Sq,p − )2 (LCoJ,(1) − LCoJ )(Sq,p ) (˜ σ (1) (r, q)Sq,p )2 (RCoJ,(1) − RCoJ )(Sq,p ) (1)



+ ΔXSq,p σ ˜ (2) (r, q)Sq,p − ρ˜(r, q)Sq,p −

+σ ˜ (2) (r, q)Sq,p ρ˜(r, q)Sq,p

$

1/2

$

(2)

(1),−

LCoJ (Sq,p )USp (1),+

RCoJ (Sp )USq,p

+ (˜ σ (2) (r, q)Sq,p − )2 (1 − (˜ ρ(r, q)Sq,p − )2 )LCoJ (Sq,p )



USq,p

+ (˜ σ (2) (r, q)Sq,p )2 (1 − (˜ ρ(r, q)Sq,p )2 )RCoJ (Sq,p )

+ (˜ σ (2) (r, q)Sq,p − )2 (LCoJ,(2) − LCoJ )(Sq,p ) + (˜ σ (2) (r, q)Sq,p )2 (RCoJ,(2) − RCoJ )(Sq,p )

1/2

1/2

(4)

USq,p

(3)

USq,p

 .

where Sq,p , p ∈ N, denotes an enumeration of the jump times of N (q). On the set Ω(n, q, r) where any two jump times Sq,p ,Sq,p in [0, T ] are further apart than 4|πn |T from each other and from all j2−r with 1 ≤ j ≤ T 2r  we have

 √  q, r)1Ω(n,q,r) = 1Ω(n,q,r) 4 n R(n, ΔN (2) (q)Sq,p ΔN (1) (q)Sq,p Sq,p ≤T

9.2 Null Hypothesis of Joint Jumps





(1)

× ΔN (2) (q)Sp σ ˜Sq,p − (r, q)Δ



289

(1)

(1)

(1) in (Sq,p )−1,n

W (1) + σ ˜Sq,p (r)Δ

(1) (1) in (Sq,p )+1,n

W (1)



(2)

+ ΔN (1) (q)Sq,p σ ˜Sq,p − (r, q)



× ρ˜Sq,p − (r, q)Δ

$

+

(2) (2)

in (Sq,p )−1,n

1 − ρ˜Sq,p − (r, q)2 Δ



(2)

+σ ˜Sq,p (r, q) ρ˜Sq,p (r, q)Δ

$

+

W (1) (2) (2)

in (Sq,p )−1,n

(2) (2)

in (Sq,p )+1,n

1 − (˜ ρSq,p (r, q))2 Δ

W (2)



W (1)

(2) (2) in (Sq,p )+1,n

W (2)



where the factor 4 stems from the fact that any jump of N (l) (q) is observed in two (l) (l) consecutive intervals Ii−1,2,n , Ii,2,n , l = 1, 2. Similarly as in the proof of (3.25) it can be shown that Condition 9.18 yields the X -stable convergence



Δ

(1) (1)

in (Sq,p )−1,n

Δ

(2) (2)

in (Sq,p )+1,n L−s

−→

$ $ $

(1) (1)

in (Sq,p )+1,n

W (1) , Δ

$ $

(i),−

W (1) , Δ

W (1) , Δ

(2) (2)

in (Sq,p )−1,n (1),−

LCoJ (Sq,p )USq,p + (1),+

RCoJ (Sq,p )USq,p + (1),−

LCoJ (Sq,p )USq,p + (1),+

RCoJ (Sq,p )USq,p +

(2)

W (2) , Δ

$

(2)

in (Sq,p )−1,n (2) (2)

in (Sq,p )+1,n

W (1) , W (2)



 Sq,p ≤T

(2),−

(LCoJ,(1) − LCoJ )(Sq,p )USq,p ,

$ $

(4),−

(3),−

(LCoJ,(2) − LCoJ )(Sq,p )USq,p ,

$

LCoJ,(2) (Sq,p )USq,p ,

(2),+

(RCoJ,(1) − RCoJ )(Sq,p )USq,p ,

(3),+

(RCoJ,(2) − RCoJ )(Sq,p )USq,p ,

$

(4),+

RCoJ,(2) (Sq,p )USq,p





Sq,p ≤T

(9.54)

(i),+

where Us , Us for i = 1, . . . , 4 are i.i.d. standard normal distributed random variables which are independent of F and of the random vectors (LCoJ,(1) , RCoJ,(1) , LCoJ,(2) , RCoJ,(2) , LCoJ , RCoJ )(s). Then (9.54) together with Lemma B.3 and the continuous mapping theorem for stable convergence stated in Lemma B.5 yields  √ ΔN (2) (q)Sq,p ΔN (1) (q)Sq,p 4 n Sq,p ∈P (q,T )





(1)

× ΔN (2) (q)Sp σ ˜Sq,p − (r, q)Δ

(1) (1)

in (Sq,p )−1,n

W (1)

290

9 Testing for the Presence of Common Jumps (1)

+ ΔN (1) (q)Sq,p



+σ ˜Sq,p (r, q)Δ

(1)

in (Sq,p )+1,n

 (2) (2) σ ˜Sq,p − (r, q) ρ˜Sq,p − (r, q)Δ (2) i (S $

+



(2)

(1)

+σ ˜Sq,p (r, q) ρ˜Sq,p Δ

n

(2)

in (Sq,p )+1,n

$

+

q,p )−1,n

1 − (˜ ρSq,p − (r, q))2 Δ

(2)

W (1)



W (1)

(2) (2)

in (Sq,p )−1,n

W (2)



W (1)

1 − ρ˜Sq,p (r, q))2 Δ

(2) (2) in (Sq,p )+1,n

W (2)



L−s

−→ ΦCoJ (q, r). T

Because of P(Ω(n, q, r)) → 1 as n → ∞ for any q, r this yields (9.53). Step 3. Finally we consider

(|ΦCoJ lim lim sup P − ΦCoJ (q, r)| > ε) = 0 T T

(9.55)

q→∞ r→∞

for all ε > 0 which can be proven using that the first moments of ΓCoJ (·, dy) are uniformly bounded together with the boundedness of the jump sizes of X respectively N (q). Step 4. Combining (9.52), (9.53) and (9.55) under the use of Lemma B.6 yields (9.51). Proof of (9.52). Step 1. We first prove  √  lim P(| n V (f(2,2) , [2], πn )T − 4V (f(2,2) , πn )T − R(n)| > ε) = 0 (9.56) n→∞

for all ε > 0 where R(n) = R(1) (n) + R(2) (n) and R(l) (n) =

√



n

(l)

(3−l)

i,j≥2:ti,n ∧tj,n

(l)



(l)

(3−l)

2Δi−1,1,n X (l) Δi,1,n X (l) Δj,2,n X (3−l) ≤T

2

1{I (l)

(3−l) i,2,n ∩Ij,2,n =∅}

l = 1, 2. Therefore note that it holds  √  n V (f(2,2) , [2], πn )T − 4V (f(2,2) , πn )T   2  (1) √ (1) = n Δi−1,1,n X (1) + Δi,1,n X (1) (1)

(2)

i,j≥2:ti,n ∧tj,n ≤T

−





(2)



(1)

(2)

× Δj−1,1,n X (2) + Δj,1,n X (2)

l1 ,l2 =0,1

Δi−l1 ,1,n X (1)

× 1{I (1)

(2) i,2,n ∩Ij,2,n =∅}

2 

(2)

2

Δj−l2 ,1,n X (2)

2

1{I (1)

(2) i−l1 ,1,n ∩Ij−l2 ,1,n =∅}



,

9.2 Null Hypothesis of Joint Jumps √ + OP ( n|πn |T )  √ = n (1)



(1)

(1)

(1)





l1 ,l2 =0,1

(2)

(2)

2

(2)

+ 2 Δi−1,1,n X (1) + Δi,1,n X (1) +



(1)

2Δi−1,1,n X (1) Δi,1,n X (1) Δj−1,1,n X (2) + Δj,1,n X (2)

i,j≥2:ti,n ∧tj,n ≤T



291

2 

(1)

Δi−l1 ,1,n X (1)

2

(2)

(2)

Δj−1,1,n X (2) Δj,1,n X (2)

(2)

Δj−l2 ,1,n X (2)

2

1{I (1)

i−l1

 (2)

,1,n ∩Ij−l

2

,1,n =∅}

× 1{I (1) ∩I (2) =∅} i,2,n j,2,n √ + OP ( n|πn |T ),

(9.57)

√

where the OP ( n|πn |T )-terms are due to boundary effects. Because of the indicator 1{I (1) we may use iterated expectations and inequality (1.12) to (2) ∩I =∅} i−l1 ,1,n

j−l2 ,1,n

obtain the following upper bound for the S-conditional expectation of the sum (1) (2) over the terms (Δi−l1 ,1,n X (1) )2 (Δj−l2 ,1,n X (2) )2 in (9.57)



√ 4K n

(1)

(1)

[2],n

(2)

|Ii,2,n ||Ij,2,n |1{I (1)

(2)

(2) i,2,n ∩Ij,2,n =∅}

i,j≥2:ti,n ∧tj,n ≤T

=

32KG2,2 (T ) √ n [2],n

which converges to zero in probability as n → ∞ because of G2,2 (T ) = OP (1) by assumption. Hence we obtain (9.56) by Lemma 2.15. Step 2. Next, we will show lim lim sup P(|R(l) (n) − R(l) (n, q)| > ε) → 0

(9.58)

q→∞ n→∞

for all ε > 0 and l = 1, 2 with R(l) (n, q) =

√

n

 (l)

(3−l)

i,j≥2:ti,n ∧tj,n

(l)

(l)



(3−l)

2Δi−1,1,n X (l) Δi,1,n X (l) Δj,2,n N (3−l) (q) ≤T

× 1{I (l)

(3−l) i,2,n ∩Ij,2,n =∅}

l = 1, 2. (l) Ii,k1 ,n

(3−l,l)

Denote by Δ(j,i),(k

(3−l) ∩ Ij,k2 ,n

2

,

X the increment of X over the interval

2 ,k1 ),n (3−l\l) Δ(j,i),(k ,k ),n X 1 2

(3−l)

(l)

and by the increment of X over Ij,k1 ,n \ Ii,k2 ,n (which might be the sum of the increments over two separate intervals). Using the elementary inequality |a2 − b2 | = (ρ|a + b|)(ρ−1 |a − b|) ≤ ρ2 (a + b)2 + ρ−2 (a − b)2

292

9 Testing for the Presence of Common Jumps (2)

(2)

which holds for any a, b ∈ R and ρ > 0 with a = Δj,2,n X (2) and b = Δj,2,n N (2) (q) yields E[|R(l) (n) − R(l) (n, q)||S]    (l) √ (l) ≤2 n ρ2 E |Δi−1,1,n X (l) Δi,1,n X (l) | (l)

(3−l)

i,j≥2:ti,n ∧tj,n

≤T

 (3−l) 2   × Δj,2,n (X (3−l) + N (3−l) (q)) S  (l)  (3−l) 2    (l) + ρ−2 E |Δi−1,1,n X (l) Δi,1,n X (l) | Δj,2,n (X (3−l) − N (3−l) (q)) S

× 1{I (l)

(3−l) i,2,n ∩Ij,2,n =∅}

.

(9.59)

Using (l)

(3−l,l)

(l\(3−l))

|Δι,1,n X (l) | ≤ |Δ(j,ι),(2,1),n X (l) | + |Δ(ι,j),(1,2),n X (l) |, ι = i − 1, i,



(3−l)

Δj,2,n Y

2



(3−l,l)

≤ 3 Δ(j,i),(2,1),n Y

2



(3−l,l)

+ 3 Δ(j,i−1),(2,1),n Y

2



(3−l\l)

+ 3 Δ(j,i),(2,2),n Y

2

,

for Y = X (3−l) + N (3−l) (q) to treat increments over overlapping and nonoverlapping intervals differently we obtain that the first summand in (9.59) is bounded by

*



√ 6ρ2 n

(l)

(3−l)

i,j≥2:ti,n ∧tj,n

(l) (l) E |Δi−1,1,n X (l) Δi,1,n X (l) |

≤T

  



2 (3−l\l) × Δ(j,i),(2,2),n (X (3−l) + N (3−l) (q)) S

+

  (l) (3−l,l) E |Δ2i−1−ι,1,n X (l) Δ(j,ι),(2,1),n X (l) | ι=i−1,i

  



2 (3−l,l) × Δ(j,ι),(2,1),n (X (3−l) + N (3−l) (q)) S (l)

(l\3−l)

+ E |Δ2i−1−ι,1,n X (l) Δ(ι,j),(1,2),n X (l) |

  



2 (3−l,l) × Δ(j,ι),(2,1),n (X (3−l) + N (3−l) (q)) S

× 1{I (l)

(3−l) i,2,n ∩Ij,2,n =∅}



√ ≤ Kρ2 n

(l)

(3−l)

i,j≥2:ti,n ∧tj,n

+

  ι=i−1,i

*

(l)

(l)

(3−l)

(3−l)

(|Ii−1,1,n ||Ii,1,n |)1/2 (|Ij,2,n | + Kq |Ij,2,n |2 )

≤T

(l) (l) |I2i−1−ι,1,n |1/2 |Iι,1,n

(3−l)

∩ Ij,2,n |1/2

+

9.2 Null Hypothesis of Joint Jumps

293 (l)

(3−l)

(l)

(3−l)

× (|Iι,1,n ∩ Ij,2,n | + Kq |Iι,1,n ∩ Ij,2,n |4 )1/2 (l)

(l)

(l)

(3−l)

(l)

(3−l)

+ (|Ii−1,1,n ||Ii,1,n |)1/2 (|Iι,1,n ∩ Ij,2,n | + Kq |Iι,1,n ∩ Ij,2,n |2 )



+

× 1{I (l)

(3−l) i,2,n ∩Ij,2,n =∅}

√ [2],n ≤ Kρ2 (1 + Kq |πn |T )G2,2 (T )/ n √ + Kρ2 (1 + Kq (|πn |T )3 )1/2 n

 (l)

(l)

(l)

(3−l)

|Ii,2,n |1/2 |Ii,2,n ∩ Ij,2,n |

(3−l)

i,j≥2:ti,n ∧tj,n

≤T

× 1{I (l)

(9.60)

(3−l) i,2,n ∩Ij,2,n =∅}

where we used iterated expectations, the Cauchy-Schwarz inequality for the second summand and Lemma 1.4 repeatedly. The first term in the last bound vanishes as [2],n n → ∞ because of G2,2 (T ) = OP (1) while the second term vanishes after letting √ [2],n n → ∞, q → ∞ and then ρ → 0 by n|πn |T = OP (1) and G2,2 (T ) = OP (1) because of √



n

(l) (3−l) i,j≥2:ti,n ∧tj,n ≤T

≤

√

n



(l)

(l)

(3−l)

(l)

(l)



|Ii,2,n |1/2 |Ii,2,n ∩ Ij,2,n |1{I (l) |Ii,2,n |1/2 |Ii,2,n | ≤ nT

(l)

(3−l) i,2,n ∩Ij,2,n =∅}



(l)

|Ii,2,n |2

1/2

(l)

i≥2:ti,n ≤T

√ [2],n ≤ (23 T G2,2 (T ))1/2 + OP ( n|πn |T )

i≥2:ti,n ≤T

(9.61)

where the inequality in the second to last line follows from the Cauchy-Schwarz inequality for sums. Analogously we can bound the second summand in (9.59) by √ [2],n Kρ−2 (Kq |πn |T + 1 + eq )G2,2 (T )/ n



+ Kρ−2 (Kq (|πn |T )3 + |πn |T + eq )1/2 nT



(l)

|Ii,2,n |2

1/2

(l)

i≥2:ti,n ≤T

√

+ OP ( n|πn |T ) where the first term vanishes as n → ∞ and the second term vanishes as n → ∞ and then q → ∞ for any ρ > 0. Hence using Lemma 2.15 we have proved (9.58). Step 3. Further we will show

(l) (n, q)| > ε) → 0 lim lim sup P(|R(l) (n, q) − R

q→∞ n→∞

(9.62)

294

9 Testing for the Presence of Common Jumps

for all ε > 0 and l = 1, 2 with

(l) (n, q) = R

√



n



(l) (3−l) i,j≥2:ti,n ∧tj,n ≤T

(l)

(l)

(l)

2 Δi−1,1,n N (l) (q)Δi,1,n C (l)



(l)

2

(3−l)

+ Δi−1,1,n C (l) Δi,1,n N (l) (q) Δj,2,n N (3−l) (q) 1{I (l)

(3−l) i,2,n ∩Ij,2,n =∅}

.

Using

 (3−l,l) 2 (3−l,l) E[Δ(j,ι),(2,1),n (X (l) − N (l) (q)) Δ(j,ι),(2,1),n N (3−l) (q) |S] (l)

(3−l)

(l)

(3−l)

≤ (Kq |Iι,1,n ∩ Ij,2,n |1/6 + (eq )1/2 )|Iι,1,n ∩ Ij,2,n | which can be derived as in (8.30) we get analogously to (9.60)

E

√



n

(l)

(l)

(l)

2Δi−1,1,n (X (l) − N (l) (q))Δi,1,n (X (l) − N (l) (q))

(3−l)

i,j≥2:ti,n ∧tj,n



≤T

2

(3−l)

× Δj,2,n N (3−l) (q) 1{I (l)





√ ≤K n

(l)

(3−l)

i,j≥2:ti,n ∧tj,n

(l)

 

  S

(l)

((Kq |Ii−1,1,n | + 1 + eq )|Ii−1,1,n |)1/2

≤T (l)

+

(3−l)

i,2,n ∩Ij,2,n =∅}

(l)

(3−l)

(3−l)

× ((Kq |Ii,1,n | + 1 + eq )|Ii,1,n |)1/2 (1 + Kq |Ij,2,n |)|Ij,2,n | (l)

(l)

((Kq |I2i−1−ι,1,n | + 1 + eq )|I2i−1−ι,1,n |)1/2

ι=i−1,i (l)

(3−l)

(l)

(3−l)

× (Kq |Iι,1,n ∩ Ij,2,n |1/6 + (eq )1/2 )|Iι,1,n ∩ Ij,2,n | (l)

(l)

(l)

(l)

+ ((Kq |I2i−1−ι,1,n | + 1 + eq )|I2i−1−ι,1,n |)1/2 ((Kq |Iι,1,n | + 1 + eq )|Iι,1,n |)1/2 (l)

(3−l)

(l)

(3−l)

× (1 + Kq |Iι,1,n ∩ Ij,2,n |)|Iι,1,n ∩ Ij,2,n | [2],n

√

≤ K(Kq |πn |T + 1 + eq )(1 + Kq |πn |T )G2,2 (T )/ n



1{I (l)

(3−l) i,2,n ∩Ij,2,n =∅}





+ (Kq |πn |T + 1 + eq )1/2 (Kq (|πn |T )1/6 + (eq )1/2 ) nT

(l)

|Ii,2,n |2

1/2

(l)

i≥2:ti,n ≤T

(9.63) which vanishes as n → ∞ and then q → ∞. Furthermore we get also analogously to (9.60)

E

√

n

 (l) (3−l) i,j≥2:ti,n ∧tj,n ≤T

(l)

(l)

2Δi−1,1,n (B (l) (q) + M (l) (q))Δi,1,n (N (l) (q))

9.2 Null Hypothesis of Joint Jumps



295

2

(3−l)

× Δj,2,n N (3−l) (q) 1{I (l)





√ ≤K n

(l)

(3−l)

i,j≥2:ti,n ∧tj,n

(3−l)

i,2,n ∩Ij,2,n =∅}

(l)

  S

(l)

((Kq |Ii−1,1,n | + eq )|Ii−1,1,n |)1/2

≤T (l)

(l)

(3−l)

(3−l)

× ((1 + Kq |Ii,1,n |)|Ii,1,n |)1/2 (1 + Kq |Ij,2,n |)|Ij,2,n | (l)

(l)

(l)

(3−l)

(l)

(3−l)

+ ((Kq |Ii−1,1,n | + eq )|Ii−1,1,n |)1/2 ((1 + Kq |Ii,1,n ∩ Ij,2,n |)|Ii,1,n ∩ Ij,2,n |)1/2 (l)

(3−l)

(l)

(3−l)

× ((1 + Kq |Ii,1,n ∩ Ij,2,n |3 )|Ii,1,n ∩ Ij,2,n |)1/2 (l)

(3−l)

(l)

(3−l)

(l)

(l)

+ ((Kq |Ii−1,1,n ∩ Ij,2,n | + eq )|Ii−1,1,n ∩ Ij,2,n |)1/2 ((1 + Kq |Ii,1,n |)|Ii,1,n |)1/2 (l)

(3−l)

(l)

(3−l)

× ((1 + Kq |Ii−1,1,n ∩ Ij,2,n |3 )|Ii−1,1,n ∩ Ij,2,n |)1/2 (l)

(l)

(l)

(l)

+ ((Kq |Ii−1,1,n | + eq )|Ii−1,1,n |)1/2 ((1 + Kq |Ii,1,n |)|Ii,1,n |)1/2 (l)

(3−l)

(l)

(3−l)

× (1 + Kq |Ii,1,n ∩ Ij,2,n |)|Ii,1,n ∩ Ij,2,n | (l)

(l)

(l)

(l)

+ ((Kq |Ii−1,1,n | + eq )|Ii−1,1,n |)1/2 ((1 + Kq |Ii,1,n |)|Ii,1,n |)1/2 (l)

(3−l)

(l)

(3−l)



× (1 + Kq |Ii−1,1,n ∩ Ij,2,n |)|Ii−1,1,n ∩ Ij,2,n | 1{I (l) [2],n

≤ K(Kq |πn |T + eq )1/2 (1 + Kq |πn |T )3/2 G2,2 (T )/ n



+ (Kq |πn |T + eq )1/2 (1 + Kq |πn |T ) nT



(3−l) i,2,n ∩Ij,2,n =∅}

√

(l)

|Ii,2,n |2

1/2

(9.64)

(l)

i≥2:ti,n ≤T

which vanishes as first n → ∞ and then q → ∞. The same obviously also holds if we switch the roles of i − 1 and i. Hence using Lemma 2.15 the estimates (9.63) (l) (l) and (9.64) yield (9.62) because Δi−1,1,n N (l) (q)Δi,1,n N (l) (q) = 0 holds eventually (l)

for all i with ti,n ≤ T as there are only finitely many big jumps. Step 4. Finally it remains to show

 q) − R(n,  q, r)| > ε) = 0 lim lim sup lim sup P(|R(n,

q→∞ r→∞

(9.65)

n→∞

 q) = R (1) (n, q) + R (2) (n, q). However, the proof for (9.65) for all ε > 0 with R(n, is identical to the proof of (8.32) because we have (l)

(3−l)

|Δi−ι,1,n N (l) (q)|(Δj,2,n N (3−l) (q))2 1{I (l) (l)

(3−l)

(3−l) i,2,n ∩Ij,2,n =∅}

1Ω(n,q) (l)

= |Δi−ι,1,n N (l) (q)|(Δi−ι,1,n N (3−l) (q))2 1Ω(n,q) ≤ KΔi−ι,1,n N (q)3 for ι = 0, 1 and l = 1, 2 on the set Ω(n, q) where two different jump times of N (q) are further apart than 4|πn |T and because of P(Ω(n, q)) → 1 for any q > 0. Step 5. Combining (9.56), (9.58), (9.62) and (9.65) then yields (9.52).

296

9 Testing for the Presence of Common Jumps

For the proof of (9.46), we require the following Proposition. Proposition 9.27. Suppose Condition 9.23 is fulfilled. Then it holds

 P

Mn   1   1{Φ  CoJ

Mn



2,T ,n,m ≤Υ}

m=1





CoJ (ΦCoJ  −P → 0 (9.66) 2,T ≤ Υ|X ) > ε ∩ ΩT

for any X -measurable random variable Υ and all ε > 0. Proof. Step 1. Denote by Sq,p , p = N, an increasing sequence of stopping times which exhausts the jump times of N (q). Similarly as in the proof of Proposition 8.15 we define Y (P, n, m) = 4

P 

(1)

(2)

Δip ,n X (1) Δjp ,n X (2)

p=1

× 1{|Δ(1)



ip ,n X

1{|Δ(2) X (2) |>β|I (2) | } jp ,n jp ,n $ (1,2),− %CoJ , −) L n,m (τip ,jp ,n )U

(1) |>β|I (1) | } ip ,n

(2)



(1)

× Δjp ,n X (2) σ ˜ (1) (tip ,n (1)

+σ ˜ (1) (tip ,n , +)



$

n,(ip ,jp ),m

(1,2),+ p ,jp ),m

% CoJ R n,m (τip ,jp ,n )Un,(i

(1)

%n,m + (˜ σ (1) (tip ,n , −))2 (L

CoJ,(1)

(1)

%CoJ −L n,m )(τip ,jp ,n )

% n,m + (˜ σ (1) (tip ,n , +))2 (R 

(1)

CoJ,(1)

(2)

% CoJ −R n,m )(τip ,jp ,n )

+ Δip ,n X (1) σ ˜ (2) (tjp ,n , −)˜ ρ(τip ,jp ,n , −) (2)

+σ ˜ (2) (tjp ,n , +)˜ ρ(τip ,jp ,n , +)



$

$

1/2

(1,2),− p ,jp ),m

%CoJ L n,m (τip ,jp ,n )Un,(i (1,2),+ p ,jp ),m

% CoJ R n,m (τip ,jp ,n )Un,(i

(2)

%CoJ + (˜ σ (2) (tjp ,n , −))2 (1 − (˜ ρ(τip ,jp ,n , −))2 )L n,m (τip ,jp ,n ) (2)





(1)

Un,ip ,m

% CoJ + (˜ σ (2) (tjp ,n , +))2 (1 − (˜ ρ(τip ,jp ,n , +))2 )R n,m (τip ,jp ,n ) (2)

%n,m + (˜ σ (2) (tjp ,n , −))2 (L

CoJ,(2)

(2)

CoJ,(2)

% CoJ −R n,m )(τip ,jp ,n )

1/2

(3)

Un,jp ,m

(2)

Un,jp ,m

%CoJ −L n,m )(τip ,jp ,n )

% n,m + (˜ σ (2) (tjp ,n , +))2 R × 1{Sq,p ≤T }

1/2



9.2 Null Hypothesis of Joint Jumps (1)

297

(2)

where (ip , jp ) = (in (Sq,p ), in (Sq,p )) and further set Y (P ) = 4 ×

P 

(1)

p=1 (2) ΔXSq,p





(2)

ΔXSq,p ΔXSq,p 1{Sq,p ≤T } (1)



σSq,p −

(1),−

(1)

LCoJ (Sq,p )USq,p + σSq,p



(1),+

RCoJ (Sq,p )USq,p

 (1) (1) (2) (σSq,p − )2 (LCoJ,(1) − LCoJ )(Sq,p ) + (σSq,p )2 (RCoJ,(1) − RCoJ )(Sq,p )USq,p    (1) (2) (1),− (2) (1),+ + ΔXSq,p σSq,p − ρSq,p − LCoJ (Sq,p )USq,p + σSq,p ρSq,p RCoJ (Sq,p )USq,p  (2) (2) (3) + (σSq,p − )2 (1 − (ρSq,p − )2 )LCoJ (Sq,p ) + (σSq,p )2 (1 − (ρSq,p )2 )RCoJ (Sq,p )USq,p   (2) (2) (4) + (σSq,p − )2 (LCoJ,(2) − LCoJ )(Sq,p ) + (σSq,p )2 (RCoJ,(2) − RCoJ )(Sq,p )USq,p .

+

Using this notation we obtain by applying Lemma (4.9) similarly as in Step 1 in the proof of Proposition 4.11

 lim P

n→∞

Mn    1 

 (Y (P ) ≤ Υ|X ) > ε ∩ ΩCoJ  1{Y (P,n,m)≤Υ} − P =0 T

Mn

m=1

(9.67) for any P ∈ N. Here, Condition 9.23(iii) is needed such that Corollary 6.4 yields the consistency of the estimators for σ (1) , σ (2) and ρ. Step 2. Next we prove lim lim sup

P →∞ n→∞

Mn 1  % CoJ P(|Y (P, n, m) − Φ 2,T,n,m | > ε) = 0 Mn

(9.68)

m=1

for all ε > 0. On the set Ω(q, P, n) on which there are at most P jumps of N (q) in [0, T ] and on which two different jumps of N (q) are further apart than 2|πn |T it holds

 

 % CoJ E |Y (P, n, m) − Φ 2,T,n,m |1Ω(q,P,n) F   (l) (l) ≤K

|Δi,n (X

l=1,2 i,j:t(l) ∨t(3−l) ≤T

×



i,n

j,n

(3−l) |Δj,n X (3−l) |˜ σ (l) (n, i)E

$

(3−l)

− N (l) (q))||Δj,n X (3−l) |

%n,m (L

CoJ,(l)

% n,m +R

CoJ,(l)

(l)

 

)(ti,n )S

(l)

+ |Δi,n (X (l) − N (l) (q))|˜ σ (3−l) (n, j) ×E

$

%n,m (L

CoJ,(3−l)

% n,m +R

CoJ,(3−l)

(3−l)

 

)(tj,n )S

× 1{|Δ(l) X (l) |>β|I (l) | ∧|Δ(3−l) X (3−l) |>β|I (3−l) | } 1{I (l) ∩I (3−l) =∅} (9.69) i,n

i,n

j,n

j,n

i,n

j,n

298

9 Testing for the Presence of Common Jumps

with



(l)

(l)

(l)

(l)

σn (ti,n , −))2 + (˜ σn (ti,n , +))2 σ ˜ (l) (n, i) = (˜

1/2

.

Because of P(Ω(q, P, n)) → ∞ as n, P → ∞ for all q > 0 it suffices to show that (9.69) vanishes as first n → ∞ and then q → ∞ for proving (9.68). Reconsidering the notation introduced in Step 2 in the proof of (9.52) we define the following terms (l)

Y(i,j),n = (l)

Y(i,j),n =

1 bn

1 bn

 (l)

(l)

(l,3−l)

(Δ(k,j),(1,1),n X (l) )2

1/2



(l)

,

(l)

k =i:Ik,n ⊂[ti,n −bn ,ti,n +bn ]

(l)

(l\3−l)

(Δ(k,j),(1,1),n , X (l) )2

(9.70)

1/2 ,

(l)

k =i:Ik,n ⊂[ti,n −bn ,ti,n +bn ]

l = 1, 2. Then the Minkowski inequality yields (l)

(l)

(3−l)

(3−l)

σ ˜ (l) (n, i) ≤ Y(i,j),n + Y(i,j),n ,

(9.71)

σ ˜ (3−l) (n, j) ≤ Y(j,i),n + Y(j,i),n

which allows to treat the increments in the estimation of σ (l) , σ (3−l) over intervals (3−l) (l) which do overlap with Ij,n , Ii,n and those which do not, separately. First we derive using the Cauchy-Schwarz inequality, iterated expectations and Lemma 1.4 the following bound for (9.69) where we replaced the estimators for σ (l) , σ (3−l) with the first summands from (9.71)



K (l)

(l) (3−l) (l) E |Δi,n (X (l) − N (l) (q))||Δj,n X (3−l) |2 Y(i,j),n

(3−l)

i,j:ti,n ∨tj,n

≤T

(3−l) (l) + |Δj,n X (3−l) ||Δi,n (X (l)

(3−l) 

− N (l) (q))|2 Y(j,i),n S



2n|πn |T

× 1{I (l) ∩I (3−l) =∅} i,n



≤K (l)



(3−l)

i,j:ti,n ∨tj,n





j,n

≤T (3−l)

× K|Ij,n (3−l)

+ (K|Ij,n

×

(l)

(l)

(l)

(3−l)

(Kq |Ii,n |2 + |Ii,n | + eq |Ii,n |)(|Ij,n

(l)

|

|/bn )

1/2

1/2

|)(|Ii,n |/bn )

1/2 



(l)

(l)

n|πn |T 1{I (l) ∩I (3−l) =∅} i,n

(l)

Kq |Ii,n |4 + |Ii,n |2 + eq |Ii,n |

j,n

≤ K(Kq (|πn |T + (|πn |T )3 ) + 1 + eq )1/2 (|πn |T /bn )1/2

1/2 

9.2 Null Hypothesis of Joint Jumps ×

√



n (l)



(3−l)

i,j:ti,n ∨tj,n

299

(l)

(3−l)

|Ii,n |1/2 |Ij,n

(3−l) 1/2 

(l)

| + |Ii,n ||Ij,n

|

1{I (l) ∩I (3−l) =∅} i,n

≤T

j,n

(9.72) for l = 1, 2 which vanishes as n → ∞ for any q > 0 because of Condition 9.23(ii) P

and because of |πn |T /bn −→ 0 by assumption. Further we obtain the following bound for the S-conditional expectation of (9.69) where we replaced the estimators for σ (l) , σ (3−l) with the second summands from (l) (3−l) (9.71) by treating increments over Ii,n ∩Ij,n and increments over non-overlapping (l)

(3−l)

parts of Ii,n , Ij,n



K (l)

(3−l)

i,j:ti,n ∨tj,n



differently (l\3−l)

(3−l)

(l)

E[|Δ(i,j),(1,1),n (X (l) − N (l) (q))||Δj,n X (3−l) |2 Y(i,j),n

≤T

(l,3−l) + |Δ(i,j),(1,1),n (X (l)

(3−l\l)

(l)

− N (l) (q))||Δ(j,i),(1,1),n X (3−l) |2 Y(j,i),n |S]

(3−l\l)

(l)

(3−l)

+ E[|Δ(j,i),(1,1),n X (3−l) ||Δi,n (X (l) − N (l) (q))|2 Y(i,j),n (3−l,l)

(l\3−l)

(3−l)

+ |Δ(j,i),(1,1),n X (3−l) ||Δ(i,j),(1,1),n (X (l) − N (l) (q))|2 Y(j,i),n |S] (l,3−l) + E[|Δ(i,j),(1,1),n (X (l)



n|πn |T

(l,3−l) (l) − N (l) (q))||Δ(i,j),(1,1),n X (3−l) |2 Y(i,j),n |S]

× E[

$

(l)

(l)

(l)

(3−l)

%n,m + R % n,m )(t ∧ t (L i,n j,n )|S] 

(l,3−l) (l,3−l) (3−l) + E[|Δ(i,j),(1,1),n X (3−l) ||Δ(i,j),(1,1),n (X (l) − N (l) (q)|2 Y(j,i),n |S]

× E[



≤K (l)



(3−l)

i,j:ti,n ∨tj,n

+

(l) ((Kq |Ii,n

(3−l)

(3−l)

(3−l) ∩ Ij,n |2

(l)

(l) + |Ii,n

(3−l) 2/3

(3−l) 1/2

+ |Ii,n ∩ Ij,n



(3−l)

i,n

| + 1 + eq )|Ii,n

(l)

(l)

j,n

(l) (l) (3−l) (Kq |Ii,n | + 1 + eq )|Ii,n |)1/2 |Ij,n |

× |Ii,n ∩ Ij,n (l)

(3−l)

%n,m + R % n,m )(t ∧ t (L i,n j,n )|S] 1{I (l) ∩I (3−l) =∅}

≤T

(3−l)

+ (Kq |Ii,n

$

|

|

E[

(l)

|)|Ij,n |1/2

(3−l) ∩ Ij,n |1/2

$

(l)

(l)

(3−l) 1/2

|)

n|πn |T 1{I (1) ∩I (2) =∅} i,n

(l) + eq )|Ii,n (l)

(3−l) 3

E[

(l)

(3−l)

$

(l)

(3−l)

| + |Ii,n ∩ Ij,n

(3−l)

(3−l)

| + eq )

(l)

1

bn

 (l)

(3−l)



%n,m + R % n,m )(t ∧ t (L i,n j,n )|S] . (9.73)

Here, we used iterated expectations and (l) E[Y(i,j),n |S] ≤

j,n

(3−l) ∩ Ij,n |)1/3

%n,m + R % n,m )(t ∧ t (L i,n j,n )|S]

((Kq |Ii,n ∩ Ij,n

× |Ii,n ∩ Ij,n



(l)

(l)

K|Ik,n | (l)

k =i:Ik,n ⊂[ti,n −bn ,ti,n +bn ]

1/2

≤K

300

9 Testing for the Presence of Common Jumps (3−l)

along with the similar bound for E[Y(j,i),n |S]. The sum over the expression in the first set of square brackets in (9.73) vanishes just like (9.72) while the sum over the expression in the second set of square brackets is bounded by



K(Kq (|πn |T )1/2 + eq )1/3

(l)

(3−l)

|Ii,n ∩ Ij,n

|

(l) (3−l) i,j:ti,n ∨tj,n ≤T



Kn 

×

n ˜ 1 ,k ˜2 =−K k

× ((n|I

(l) |I ˜ i+k1 ,n

(3−l) ∩I ˜ | j+k2 ,n





Kn   ˜  ,k ˜ k 1 2 =−Kn

(l) (l) 1/2 ˜1 −1,n | + n|Ii+k ˜1 +1,n |) i+k

(l)

(l) ˜  ,n i+k 1

(l)

(3−l)

∩I

−1 (3−l) | ˜  ,n j+k 2

(3−l) (3−l) 1/2 ) ˜2 −1,n | + n|Ij+k ˜2 +1,n |) j+k (3−l)

|Ii,n ∩ Ij,n

i,j:ti,n ∨tj,n (l)

+ (n|I



= K(Kq (|πn |T )1/2 + eq )

|I

|

≤T

(l)

(3−l)

(3−l)

× ((n|Ii−1,n | + n|Ii+1,n |)1/2 + (n|Ij−1,n | + n|Ij+1,n |)1/2 ) 

×

Kn  n ˜2 =−K ˜ 1 ,k k

(l) |I ˜ i+k1 ,n

(3−l) ∩I ˜ | j+k2 ,n

√





Kn   ˜ ˜  ,k k 1 2 =−Kn

|I

(l) ˜  ,n ˜ 1 +k i+k 1

∩I

−1 (3−l) |  ˜ ,n ˜2 +k j+k

+ OP ( n|πn |T )

2

(9.74)

where the sum in the second to last line is less or equal than 4 which can be shown √ similarly to (4.23). The OP ( n|πn |T )-term is due to boundary effects. Hence (9.74) is bounded by    √  (l) (3−l) |Ii,2,n |3/2 + |Ij,2,n |3/2 K(Kq (|πn |T )1/2 + eq ) n (l)

(3−l)

i≥2:ti,n ≤T

√

j≥2:tj,n

≤T

+ OP ( n|πn |T ) which vanishes as shown in (9.61) for first n → ∞ and then q → ∞ because of [2],n G2,2 (T ) = OP (1). Step 3. Further we obtain P  (Y (P ) ≤ Υ|X )1 CoJ −→ P P(ΦCoJ 2,T ≤ Υ|X )1ΩCoJ Ω T

T

(9.75)

as in Step 3 in the proof of Proposition 8.15. Step 4. Finally, (9.66) is obtained from (9.67), (9.68) and (9.75); compare Steps 4 and 5 in the proof of Proposition 4.10. Proof of Theorem 9.24. Analogously as in the proof of (8.33) we obtain

 lim P

n→∞







CoJ CoJ %CoJ |Q =0 2,T,n (α) − Q2,T (α)| > ε ∩ ΩT

(9.76)

9.2 Null Hypothesis of Joint Jumps

301

for all ε > 0 and any α ∈ [0, 1] from Proposition 9.27. Then Theorem 9.19 and (9.76) yield, compare Step 2 in the proof of Theorem 4.3,

   CoJ    √n V (f(2,2) , [2], πn )T − 4V (f(2,2) , πn )T  > Q  %CoJ P →α 2,T,n (1 − α) F from which we immediately conclude (9.46). nCoJ For proving (9.47) we observe that ΨCoJ to a 2,T,n converges on the given F random variable which is under Condition 9.23(iv) almost surely different from 1 DisJ  DisJ k,T by Theorem 9.16. To see this observe that C , C1,T are F -measurable while

DisJ  DisJ  k,T the F -conditional distribution of D , D1,T is continuous by Condition 9.23. DisJ DisJ DisJ DisJ k,T 1,T  k,T  1,T Hence, kC = C or D = D almost surely imply DisJ DisJ DisJ DisJ k,T  k,T 1,T  1,T +D = C +D a.s. kC

Then (9.47) follows from CoJ c2,T,n (α)1ΩnCoJ = oP (1) T

(9.77)

which we will prove in the following: From the proof of Theorem 9.19 we know that nV (f(2,2) , πn )T converges on ΩnCoJ stably in law to a non-negative random T variable. Comparing (8.44) in the proof of Theorem 8.11 it then suffices to show √

% CoJ nΦ 2,T,n,m 1ΩnCoJ = oP (1)

(9.78)

T

uniformly in m for proving (9.77). In order to achieve this goal, observe that it holds √

% CoJ E[ n|Φ |F] 2,T,n,m |1ΩnCoJ T  √ 

(l)

(3−l)

|Δi,n X (l) |(Δj,n X (3−l) )2

≤K n

l=1,2 i,j:t(l) ∨t(3−l) ≤T i,n

× |˜ σ

(l)

j,n

(l) (l) (ti,n , −) + σ ˜ (l) (ti,n , +)|

%n,m × E[((L

CoJ,(l)

% n,m +R

CoJ,(l)

(l)

(3−l)

)(ti,n ∧ tj,n ))1/2 |F]1{|Δ(l) X (l) |>β|I (l) | } i,n

i,n

× 1{|Δ(3−l) X (3−l) |>β|I (3−l) | } 1{I (l) ∩I (3−l) =∅} 1ΩnCoJ √

≤K n

j,n





j,n

i,n

l=1,2 i,j:t(l) ∨t(3−l) ≤T i,n

(l)

T

j,n

(l) (l) (3−l) |Ii,n |− (Δi,n X (l) )2 (Δj,n X (3−l) )2

j,n

(l)

× |˜ σ (l) (ti,n , −) + σ ˜ (l) (ti,n , +)|



n|πn |T 1{I (l) ∩I (3−l) =∅} 1ΩnCoJ . i,n

j,n

T

(9.79)

302

9 Testing for the Presence of Common Jumps

Using the notation from (9.70) and the inequalities (9.71) we then obtain (l)

(3−l)

(l)

(l)

σ (l) (ti,n , −) + σ ˜ (l) (ti,n , +)|1ΩnCoJ (Δi,n X (l) )2 (Δj,n X (3−l) )2 |˜ T

(l)

(3−l,l)

≤ 2(Δi,n X (l) )2 (Δ(j,k),(1,1),n X (3−l) )2 ×

1



bn

(l)

(l)

(Δk,n X (l) )2

(l)

1/2

1ΩnCoJ

(l)

(3−l\l)

+ 2(Δi,n X (l) )2 (Δ(j,k),(1,1),n X (3−l) )2

1

bn

 (l)

(l,3−l)

(Δ(k,j),(1,1),n X (l) )2

(l)

1/2

1ΩnCoJ . (9.81) T

(l)

k =i:Ik,n ⊂[ti,n −bn ,ti,n +bn ]

(l)

(3−l\l)

+ 2(Δi,n X (l) )2 (Δ(j,k),(1,1),n X (3−l) )2

1

bn

(9.80)

T

(l)

k =i:Ik,n ⊂[ti,n −bn ,ti,n +bn ]

 (l)

(l\3−l)

(Δ(k,j),(1,1),n X (l) )2

(l)

1/2

1ΩnCoJ . (9.82) T

(l)

k =i:Ik,n ⊂[ti,n −bn ,ti,n +bn ]

Recall that we may write X = B(q ) + C + M (q ) for some q > 0 by Condition 1.3; compare (9.20). Then iterated expectations and Lemmata 1.4, 3.30, 9.11 yield that (l) (l) (3−l) the S-conditional expectation of (9.80) is bounded by K|Ii,n ||Ii,n ∩ Ij,n |. Furthermore (9.81) is using iterated expectations, the Cauchy-Schwarz inequality and (l) (3−l) (l) (1.12) bounded by K|Ii,n ||Ij,n \ Ii,n | and (9.82) is using iterated expectations (l)

(3−l)

(l)

and inequality (1.12) bounded by K|Ii,n ||Ij,n \ Ii,n | as well. Then alltogether we obtain that the S-conditional expectation of (9.79) is bounded by √  K n



(l)

(l)

(3−l)

|Ii,n |− |Ii,n ||Ij,n

l=1,2 i,j:t(l) ∨t(3−l) ≤T

√  =K n

i,n

|



n|πn |T 1{I (l) ∩I (3−l) =∅} i,n

j,n



(l)

(l)

(3−l)

|Ii,n |1/2− |Ii,n |1/2 |Ij,n

|

j,n



n|πn |T

l=1,2 i,j:t(l) ∨t(3−l) ≤T i,n

≤ Kn1/2 (|πn |T )

j,n

1− 

× 1{I (l) ∩I (3−l) =∅} i,n

j,n

n Gn 1,2 (T ) + G2,1 (T )



where we used 1/2 −  > 0. Hence (9.78) follows from Condition 9.23(ii) and Lemma 2.15. Proof of Corollary 9.25. Note that it holds n−1 ACoJ 2,T,n = oP (1). To see this observe that the second sum in n−1 ACoJ vanishes as n → ∞ because big common jumps 2,T,n are asymptotically filtered out due to the indicator and because the remaining

9.2 Null Hypothesis of Joint Jumps

303

terms vanish by (2.14) as well. The first sum in n−1 ACoJ 2,T,n can be discussed similarly. Hence on ΩCoJ it holds T √

n

n−1 ACoJ 2,T,n

P

4V (f(2,2) , πn )T

1ΩCoJ −→ 0 T

and combining this with (9.43) yields the X -stable convergence √  CoJ n Φ2,T,n − ρ



n−1 ACoJ 2,T,n 4V (f(2,2) , πn )T

L−s

− 1 −→

ΦCoJ 2,T

(9.83)

4B ∗ (f(2,2) )T

on ΩCoJ . Replacing (9.43) with (9.83) in the proof of (9.46) yields (9.48). T Using arguments from the proof of Theorem 9.16 and the proof of (9.10) we obtain L−s





DisJ DisJ DisJ DisJ − C 1,T  2,T  1,T ACoJ ) + 4(D −D ) 1ΩnCoJ (9.84) 2,T,n 1ΩnCoJ −→ 4(2C2,T T

T

under the alternative ω ∈ conclude



ΦCoJ 2,T,n − ρ

ΩnCoJ . T

ACoJ 2,T,n 4nV (f(2,2) , πn )T L−s

−→ (1 − ρ)

Hence based on Theorem 9.16 and (9.84) we



− 1 1ΩnCoJ T

DisJ 2,T (2C

DisJ DisJ DisJ 1,T  2,T  1,T −C ) + (D −D )

DisJ + D  DisJ C 1,T 1,T

1ΩnCoJ T

where the limit is almost surely different from zero by Condition 9.23(iv). We then CoJ obtain (9.49) as in the proof of Theorem 9.24 because of c2,T,n (α)1ΩnCoJ = oP (1); T compare (9.77).

Bibliography [1] Yacin A¨ıt-Sahalia, Jianqing Fan, and Dacheng Xiu. High-frequency estimates with noisy and asynchronous financial data. Journal of the American Statistical Association, 492(105):1504–1516, 2010. [2] Yacine A¨ıt-Sahalia and Jean Jacod. Testing for jumps in a discretely observed process. The Annals of Statistics, 37(1):184–222, 2009. [3] Yacine A¨ıt-Sahalia and Jean Jacod. High-Frequency Financial Econometrics. Princeton University Press, 2014. ISBN: 0-69116-143-3. [4] Torben G. Andersen and Tim Bollerslev. Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review, 39(4):885–905, 1998. [5] Ole Eiler Barndorff-Nielsen, Peter Reinhard Hansen, Asger Lunde, and Neil Shephard. Multivariate realised kernels: Consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading. Journal of Econometrics, 162(2):149–169, 2011. [6] Ole Eiler Barndorff-Nielsen and Neil Shephard. Econometric analysis of realized volatility and its use in estimating stochastic volatility models. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 64(2):253– 280, 2002. [7] Markus Bibinger, Nikolaus Hautsch, Peter Malec, and Markus Reiß. Estimating the spot covariation of asset prices - statistical theory and empirical evidence. To appear in: Journal of Business and Economic Statistics, 2017. [8] Markus Bibinger and Mathias Vetter. Estimating the quadratic covariation of an asynchronously observed semimartingale with jumps. Annals of the Institute of Statistical Mathematics, 67:707–743, 2015. [9] Patrick Billingsley. Convergence of probability measures. Wiley series in probability and statistics. Wiley, 2nd edition, 1999. [10] Peter J. Brockwell and Richard A. Davis. Time Series: Theory and Methods. Springer Series in Statistics. Springer, 2nd edition, 1991.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 O. Martin, High-Frequency Statistics with Asynchronous and Irregular Data, Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics, https://doi.org/10.1007/978-3-658-28418-3

306

Bibliography

[11] Donald Lyman Burkholder. Martingale transforms. The Annals of Mathematical Statistics, 37:1494–1504, 1966. [12] Marek Capinski, Ekkehard Kopp, and Janusz Traple. Stochastic Calculus for Finance. Cambridge University Press, 2012. [13] Freddy Delbaen and Walter Schachermayer. A general version of the fundamental theorem of asset pricing. Mathematische Annalen, 3(300):463–520, 1993. [14] Thomas W. Epps. Comovements in stock prices in the very short run. Journal of the American Statistical Association, 74(366):291–298, 1979. [15] Jianqing Fan and Yazhen Wang. Spot volatility estimation for high-frequency data. Statistics and its Interface, 1:279–288, 2008. [16] Thomas S. Ferguson. A course in large sample theory. Chapman & Hall, 1st edition, 1996. [17] Otto Forster. Analysis 2 - Differentialrechnung im Rn , gew¨ ohnliche Differentialgleichungen. Vieweg+Teubner, 8th edition, 2008. [18] Masaaki Fukasawa and Mathieu Rosenbaum. Central limit theorems for realized volatility under hitting times of an irregular grid. Stochastic Processes and their Applications, 122(12):3901–3920, 2012. [19] Godfrey Harold Hardy, John Edensor Littlewood, and George P´ olya. Inequalities. Cambridge Mathematical Library. Cambridge University Press, 2nd edition, 1952. [20] Takaki Hayashi, Jean Jacod, and Nakahiro Yoshida. Irregular sampling and central limit theorems for power variations: the continuous case. Annales de l’Institut Henri Poincar´e Probabilit´es et Statistiques, 47(4):1197–1218, 2011. [21] Takaki Hayashi and Shigeo Kusuoka. Consistent estimation of covariation under nonsychronicity. Statistical Inference for Stochastic Processes, 11(1):93– 106, 2008. [22] Takaki Hayashi and Nakahiro Yoshida. On covariance estimation of nonsynchronously observed diffusion processes. Bernoulli, 11(2):359–379, 2005. [23] Takaki Hayashi and Nakahiro Yoshida. Asymptotic normality of a covariance estimator for non-synchronously observed processes. Annals of the Institute of Statistical Mathematics, 60(2):367–406, 2008.

Bibliography

307

[24] Takaki Hayashi and Nakahiro Yoshida. Nonsynchronous covarianciation process and limit theorems. Stochastic Processes and their Applications, 121:2416–2454, 2011. [25] Xin Huang and George Tauchen. The relative contribution of jumps to total price variance. J. Financial Econometrics, 4:456–499, 2006. [26] Jean Jacod. On continuous conditional gaussian martingales and stable convergence in law. S´eminaire de Probabilit´es, XXXI:232–246, 1997. [27] Jean Jacod. Asymptotic properties of realized power variations and related functionals of semimartingales. Stochastic Processes and their Applications, 118(4):517–559, 2008. [28] Jean Jacod. Statistics and high frequency data, 2009. Lecture Notes, SEMSTAT Course in La Manga. [29] Jean Jacod and Philip Protter. Asymptotic error distributions for the Euler method for stochastic differential equations. The Annals of Probability, 26:267– 307, 1998. [30] Jean Jacod and Philip Protter. Discretization of Processes. Springer, 2012. ISBN: 3-64224-126-3. [31] Jean Jacod and Albert Shiryaev. Limit Theorems for Stochastic Processes. Springer, 2nd edition, 2002. ISBN: 3-540-43932-3. [32] Jean Jacod and Viktor Todorov. Testing for common arrivals of jumps for discretely observed multidimensional processes. The Annals of Statistics, 37(1):1792–1838, 2009. [33] Achim Klenke. Probability Theory: A Comprehensive Course. Springer-Verlag London, 2nd edition, 2014. [34] Yuta Koike. An estimator for the cumulative co-volatility of asynchronously observed semimartingales with jumps. Scandinavian Journal of Statistics, 2(41):460–481, 2014. [35] Cecilia Mancini and Fabio Gobbi. Identifying the brownian covariation from the co-jumps given discrete observations. Econometric Theory, 28(2):249–273, 2012. [36] Ole Martin and Mathias Vetter. Testing for simultaneous jumps in case of asynchronous observations. Bernoulli, 24(4B):3522–3567, 2018. [37] Ole Martin and Mathias Vetter. Laws of large numbers for Hayashi-Yoshidatype functionals. Finance and Stochastics, 23(3):451–500, 2019.

308

Bibliography

[38] Ole Martin and Mathias Vetter. The null hypothesis of common jumps in case of irregular and asynchronous observations. Scandinavian Journal of Statistics, to appear, 2019+. [39] Per A. Mykland and Lan Zhang. Assessment of uncertainty in high frequency data: The observed asymptotic variance. Econometrica, 37(1):197–231, 2017. [40] Mark Podolskij and Mathias Vetter. Understanding limit theorems for semimartingales: a short survey. Statistica Neerlandica, 64:329–351, 2010. [41] Philip Protter. Stochastic Integration and Differential Equations. Springer, 2nd edition, 2004. ISBN: 978-3-642-05560-7. [42] Daniel Revuz and Marc Yor. Continuous Martingales and Brownian Motion. Springer, 3rd edition, 1999. [43] Gennady Samorodnitsky and Murad Taqqu. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Stochastic Modeling Series). Chapman & Hall, 1994. [44] Carl P. Simon and Lawrence E. Blume. Mathematics for Economists. W. W. Norton & Company, 1994. [45] Roy L. Streit. Poisson Point Processes: Imaging, Tracking, and Sensing. Springer, 2010. [46] Viktor Todorov and George Tauchen. Volatility jumps. Journal of Business and Economic Statistics, 29:356–371, 2011. [47] Mathias Vetter and Tobias Zwingmann. A note on central limit theorems for quadratic variation in case of endogenous observation times. Electronic Journal of Statistics, 11(1):963–980, 2017.

Appendix

A Estimates for Itˆ o Semimartingales The following inequalities allow to bound increments of a local martingale M by increments of its quadratic variation [M, M ]. The version here is taken from (2.1.34) in [30]. An inequality of that form was first mentioned in [11] for continuous martingales. Lemma A.1 (Burkholder-Davis-Gundy inequalities). Let M be a local martingale with M0 = 0 and S ≤ S be two stopping times. Then for p ≥ 1 there exist constants 0 < cp < Cp < ∞ such that it holds cp E[([M, M ]S  ] − [M, M ]S )p/2 |FS ] ≤ E[ sup

t∈(S,S  ]

|Mt − MS |p |FS ]

≤ Cp E[([M, M ]S  ] − [M, M ]S )p/2 |FS ]. (A.1) The constants cp , Cp are universal i.e. do not depend on the local martingale M or the stopping times S, S . Proof. A proof of the inequalities can be found for example in Chapter IV §4 of [42]. Lemma A.1 is used e.g. to derive the bounds for C and M (q) in the proof of Lemma 1.4. Proof of Lemma 1.4. Using |1{δ(s,z)≤1} −1{γ(z)≤1/q} | ≤ 1{γ(z)>1∧1/q} it holds



s+t 

B(q)s+t − B(q)s  ≤



s

≤



s



s

≤ ≤

s+t  s+t  s+t 

s

 bu  +

 bu  +



R2

R2

δ(s, z)1{γ(z)>1∧1/q} λ(dz) du

Γu bu  + 1 ∧ 1/q bu  +



Γu (γ(z))2 1 λ(dz) du 1 ∧ 1/q {γ(z)>1∧1/q}





R2

(γ(z)2 ∧ K 2 )λ(dz) du

Γu (1 ∨ K 2 ) 1 ∧ 1/q





R2

(γ(z)2 ∧ 1)λ(dz) du

≤ Kq t © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 O. Martin, High-Frequency Statistics with Asynchronous and Irregular Data, Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics, https://doi.org/10.1007/978-3-658-28418-3

312

Appendix

where we used that bt , Γt are bounded by assumption and because γ ≤ K for some constant K > 0 and R2 (γ(z)2 ∧ 1)λ(dz) < ∞ by Condition 1.3. Taking this inequality to the p-th power yields (1.8). For p ≥ 1 the inequality (1.9) follows by the boundedness of σ directly from (2.1.34) in [30]. Using (1.9) for p = 1 and the Jensen inequality for the function x → x1/p , x ≥ 0, p ∈ (0, 1), we then obtain

E[Cs+t − Cs p |Fs ] ≤ (E[Cs+t − Cs |Fs ]p ≤ (K1 )p tp/2 which yields (1.9) for p ∈ (0, 1). Regarding the inequalities (1.10) and (1.11), Lemma 2.1.5 and Lemma 2.1.7 in [30] yield

    p/2  E M (q)s+t − M (q)s p |Fs ≤ Kp sE δ%M (p)s,t |Fs + sp/2 E δ%M (2)s,t |Fs , (A.2) 

        p E N (q)s+t − N (q)s p |Fs ≤ Kp sE δ%N (p )s,t |Fs + sp E δ%N (1)s,t |Fs (A.3) for p ≥ 2 and p ≥ 1 with 1 δ%M (r)s,t = t

δ%N (r)s,t =

1 t

 

s+t  s s+t 

R2

s

R2

δ(u, z)r 1{γ(z)≤1/q} λ(dz)du,

r ≥ 2,

δ(u, z)r 1{γ(z)>1/q} λ(dz)du,

r ≥ 1.

We derive for p ≥ 2 δ%M (p)s,t ≤

1 t



s+t  s

R2

≤ K(1/q)p−2

((Γu γ(z))p 1{γ(z)≤1/q} λ(dz)du



R2

(γ(z)2 ∧ 1/q 2 )1{γ(z)≤1/q} λ(dz)

≤ K(1/q)p−2 max{1, 1/q 2 }



R2

(γ(z)2 ∧ 1)1{γ(z)≤1/q} λ(dz)

which tends to zero as q → ∞. Hence (A.2) yields (1.10) for all p ≥ 2 using tp/2 ≤ T p/2−1 t. The result for p ∈ (0, 2) then follows using Jensen’s inequality as for (1.9).

A Estimates for Itˆ o semimartingales

313

Further, by Condition 1.3 there exists a constant K ≥ 0 with γ(z) ≤ K and by  assumption a constant K with (Γu )p ≤ K which yields δ%N (p )s,t ≤ K





R2



≤ K Kp ≤ K Kp for p ≥ 2 and δ%N (p )s,t ≤ K



(γ(z))p 1{γ(z)>1/q} λ(dz) −2



R2 

−2

(γ(z)2 ∧ K 2 )λ(dz)

max{1, K 2 }



 R2

(γ(z)2 ∧ 1)λ(dz)



R2

(γ(z))p 1{γ(z)>1/q} λ(dz)

2−p



≤K q

R2

(γ(z)2 ∧ K 2 )1{γ(z)>1/q} λ(dz)



≤ K q 2−p max{1, K 2 }



R2

(γ(z)2 ∧ 1)λ(dz)

for p ∈ [1, 2). Hence (A.3) yields (1.11). As γ is bounded by Condition 1.3 there exists a q > 0 with Xt = X0 + B(q )t + Ct + M (q )t . The inequality (1.12) then follows from the elementary inequality a + b + cp ≤ Kp (ap + bp + cp ), a, b, c ∈ R2 , p ≥ 0, and inequalities (1.8)–(1.11). Proof of Lemma 1.5. The proof for (1.13) is identical to the proof of (1.8) as all estimates are ω-wise. As Ct is a local martingale we obtain the following estimate from the upper Burkholder-Davis-Gundy inequality (Lemma A.1)

E[ sup Ct − CS p |FS ] t∈(S,S  ]

≤ Kp

 l=1,2

≤ Kp



(l)

t∈(S,S  ]

Kp E[([C (l) , C (l) ]S  ] − [C (l) , C (l) ]S )p/2 |FS ]

l=1,2

= Kp

 l=1,2

≤ Kp



l=1,2

(l)

E[ sup |Ct − CS |p |FS ]



S

E[(

(l)

(σs )2 ds)p/2 |FS ]

S

E[(K(S − S))p/2 |FS ] ≤ Kp E[(S − S)p/2 |FS ]

314

Appendix

which is (1.14). As Mt is also a local martingale we obtain again using Lemma A.1

E[ sup M (q)t − M (q)S 2 |FS ] t∈(S,S  ]



≤K

E[([M (l) (q), M (l) (q)]S  ] − [M (l) (q), M (l) (q)]S )|FS ]

l=1,2



=K



S

E[ S

l=1,2



S



≤ 2K E[ S  S



≤ 2K E[



S S

= 2K E[

 ≤ 2K

R2

R2

S

R2



R2

 R2

(δ (l) (s, z))2 1{γ(z)≤1/q} μ(ds, dz)|FS ]

δ(s, z)2 1{γ(z)≤1/q} μ(ds, dz)|FS ] ((Γs γ(z))2 ∧ K )1{γ(z)≤1/q} μ(ds, dz)|FS ] ((Γs γ(z))2 ∧ K )1{γ(z)≤1/q} λ(dz)ds|FS ]

((γ(z))2 ∧ 1)1{γ(z)≤1/q} λ(dz)E[(S − S)|FS ]

where we used that the quadratic variation of M (q) equals the sum of squared jumps of M (q), that (Γs γ(z))2 is bounded by some constant K and that the Lebesgue measure times λ is the predictable compensator of μ. This estimate yields (1.15) with

 eq =

R2

((γ(z))2 ∧ 1)1{γ(z)≤1/q} λ(dz).

The last inequality (1.16) follows from Xt = X0 + B(q )t + Ct + M (q )t for a sufficiently small q as the jumps of X are bounded through γ by a constant.

B Stable Convergence in Law In this appendix chapter we give an introduction to the concept of stable convergence in law and a short overview of its basic properties. Stable convergence in law is a slightly stronger mode of convergence of random variables than convergence in law, but still weaker than convergence in probability.1 To illustrate the need for a stronger mode of convergence than convergence in law we consider the following generic situation which has already been briefly skteched in the beginning of Chapter 3; compare also Section 2 in [40] and Section 2.2.1 in [30]. Let Yn , n ∈ N, be a sequence of random variables which converges in law to a mixed normal distribution i.e. it holds L

Yn −→ V U

(B.1)

for independent random variables V > 0 and U ∼ N (0, 1). In the case where (B.1) depicts a central limit theorem, the asymptotic variance V 2 has to be estimated such that the central limit theorem can be of use for statistical inference. To this end one usually first looks for a sequence of estimators Vn2 , n ∈ N, which P

consistently estimates the asymptotic variance V 2 , i.e. fulfils Vn2 −→ V 2 , and after that one tries to deduce the convergence Y

n

L

Vn2

−→ U

(B.2)

where all components in the equation, the estimators Yn , Vn and the asymptotic law of U , are known to the statistician. Hence (B.2) can be used for statistical purposes while (B.1) is infeasible as it contains the unknown variable V . To obtain P

(B.2) from (B.1) and Vn2 −→ V 2 usually the joint convergence L

(Yn , Vn2 ) −→ (U V, V 2 )

(B.3)

is used together with the continuous mapping theorem. However, (B.3) follows P

from (B.1) and Vn2 −→ V 2 in general only if V is deterministic; compare Theorem 6(c) in [16]. If V is non-deterministic this implication is in general false. Hence, some stronger requirement on the convergence of the sequences Yn and Vn is needed to obtain (B.3), which then using the continuous mapping theorem is 1 Compare

the first paragraph on page 4 of [40].

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 O. Martin, High-Frequency Statistics with Asynchronous and Irregular Data, Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics, https://doi.org/10.1007/978-3-658-28418-3

316

Appendix

sufficient to get (B.2). This can be achieved if instead of convergence in law of the random variables Yn to V U in (B.1) we require stable convergence in law which is introduced in the following definition; compare Definition 2.1 in [40]. Definition B.1. Let Yn , n ∈ N, be a squence of random variables defined on a probability space (Ω, F , P) with values in some Polish space (E, E). Further let G ⊂ F denote some sub-σ-algebra. We say that Yn converges G-stably in law to L−s

some random variable Y , written as Yn −→ Y , defined on an extended probability ) if and only if it holds  F, P space (Ω,

 [Zg(Y )] E[Zg(Yn )] −→ E

(B.4)

for any bounded, continuous function g : E → R and any bounded real-valued G-measurable random variable Z. 

) is an extension of (Ω, F , P) means that (Ω, F , P) is  F, P Here, the notion that (Ω, ) and all F -measurable random variables have  F, P in some sense contained in (Ω,  under P and P the same distribution. For a more precise definition see (2.1.26) in [30]. Remark B.2. As for ordinary convergence in law it is usually sufficient to require that (B.4) holds for smaller classes of functions g, i.e. if (B.4) holds for all Lipschitzcontinuous functions g with compact support it automatically also holds for all bounded, continuous functions.  In the context of high-frequency statistics the asymptotic variance very often depends on the observed path t → Xt (ω) or its components b, σ and δ, compare Chapers 3 and 7-9. Hence in this field the concept of stable convergence in law is very useful as the sequences Yn , n ∈ N, of interest are based on the observed (l) values X (l) of X (l) on a discrete grid and therefore depend on the observad path ti,n

as well. Consequently the asymptotic variance is very often random and dependent on the sequence Yn , n ∈ N. The following lemma states two useful equivalent characterizations of stable convergence in law; compare Proposition 2.2 in [40]. Lemma B.3. Let Y , Yn , n ∈ N, be random variables as in Definition B.1. Then the following statements are equivalent. L−s

(i) The G-stable convergence Yn −→ Y holds. L

(ii) The convergence in law (Z, Yn ) −→ (Z, Y ) holds for any G-measurable random variable Z.

B Stable convergence in law

317 L−s

(iii) The G-stable convergence (Z, Yn ) −→ (Z, Y ) holds for any G-measurable random variable Z. Proof. The implication (iii)⇒(ii) is obvious. Further (ii) yields that

 [˜ E[˜ g (Z, Yn )] −→ E g (Z, Y )] holds for any continuous and bounded function g˜. Hence (i) follows if we set g˜(z, y) = zg(y). It remains to prove (i)⇒(iii). To this end we have to show that (i) implies

 [Z g(Z

, Y )] E[Z g(Z

, Yn )] −→ E

(B.5)

for any G-measurable random variables Z , Z

where Z is bounded. By a density argument it is sufficient to show (B.5) for all functions of the form g(z, y) = 1A (z)1B (y) and (B.5) then follows from (i) if we set Z = Z 1A (Z

) in (B.4). The following remark demonstrates why it is necessary to consider an extended ) in Definition B.1 on which the limiting variable Y is  F, P probability space (Ω, defined; compare Lemma 2.3 in [40]. In fact, it turns out that F -stable convergence in law coincides with convergence in probability if we require Y to live on the same probability space (Ω, F , P) on which the sequence Yn is defined. Further for ordinary convergence in law, only the law of the limiting variable Y and not the underlying probability space where the limiting variable Y is defined is important. Contrary we observe that for stable convergence in law also the joint law of the limiting variable Y and F -measurable random variables Z is of importance. Hence Y and these random variables Z have to be defined on a common probability space. Remark B.4. Let Y , Yn , n ∈ N, be random variables as in Definition B.1, L−s

assume that the F -stable convergence Yn −→ Y holds and suppose that Y is L

F-measurable. Then Lemma B.3 yields (Y, Yn ) −→ (Y, Y ) which by the continuous L

mapping theorem, compare Theorem 2.7 in [9], implies Yn − Y −→ 0. Next as convergence in law to a constant is equivalent to convergence in probability to P

a constant, compare Theorem 6(c) in [16], we get Yn − Y −→ 0. Hence in the situation where the limit is F-measurable F -stable convergence in law is equivalent to convergence in probability.  The following lemma generalizes the continuous mapping theorem for convergence in law, compare Theorem 2.7 in [9], to the concept of stable convergence in law.

318

Appendix

Lemma B.5. Let Y , Yn , n ∈ N, be random variables as in Definition B.1. If L−s

we assume that the G-stable convergence Yn −→ Y holds, then we also have the G-stable convergence L−s

g(Yn ) −→ g(Y )

(B.6)

for any continuous function g : (E, E) → (E , E ) for some Polish space (E , E ). Proof. To prove (B.6) by Definition B.1 we need to show

 [Zh(g(Y ))] E[Zh(g(Yn ))] −→ E

(B.7)

for any G-measurable and bounded random variable Z and any continuous function L−s

h. However, Yn −→ Y immediately yields (B.7) because the function h ◦ g is continuous and bounded. The next lemma generalizes another well-known result for convergence in law, compare Theorem 6(b) in [16], to the concept of stable convergence in law. Lemma B.6. Suppose we are given a sequence of real-valued random variables (Yn )n∈N and a double sequence of real-valued random variables (Zn,k )n,k∈N both defined on a probability space (Ω, F , P) and real-valued random variables Y , Zk , ). Further assume that it  F,  P k ∈ N, defined on an extended probability space (Ω, holds lim lim sup P(|Yn − Zn,k | > ε) = 0,

k→∞ n→∞

(B.8)

(|Y − Zk | > ε) = 0 lim P

k→∞

for any ε > 0 and that for some sub-σ-algebra G ⊂ F we have the G-stable convergences L−s

Zn,k −→ Zk , k ∈ N,

(B.9)

L−s

as n → ∞. Then the G-stable convergence Yn −→ Y also holds for n → ∞. Proof. We have to check (B.4). To this end note that it holds

 [Zg(Y )]| ≤ K E  [|g(Yn ) − g(Y )|] |E[Zg(Yn )] − E 



 [|g(Zn,k ) − g(Zk )|] + E  [|g(Zk ) − g(Y )|] . ≤ K E[|g(Yn ) − g(Zn,k )|] + E We first conclude E[|g(Zn,k ) − g(Zk )|] → 0 as n → ∞ for any k ∈ N from (B.9). Further



 [|g(Zk ) − g(Y )| lim lim sup E[|g(Yn ) − g(Zn,k )|] + E

k→∞ n→∞



B Stable convergence in law

319

follows from (B.8) for all Lipschitz-continuous functions g. However, it is sufficient to prove (B.4) for such functions only because then by a density argument (B.4) already has to hold for all bounded and continuous functions; compare Remark B.2. A standard approximation technique when working with Itˆ o semimartingales is based on the idea to first show desired results for processes with finite jump activity, then let the number of jumps tend to infinity and finally to show that the desired results remain valid in the limit. To verify this method we use Lemma (B.6) in Chapters 3 and 7-9. The upcoming proposition, compare Proposition 2.5 in [40], finally states that working with stable convergence in law instead of with ordinary convergence in law allows to draw the desired conclusions sketched in the introductory example. Proposition B.7. Let Yn , Y , X be Rd -valued random variables and Vn , V be )  F, P Rd×d -valued random variables all defined on the extended probability space (Ω, d d where Yn , Vn , V are F -measurable. Further let g : R → R be a continuously differentiable function. Then: L−s

P

(i) If the X -stable convergence Yn −→ Y holds and we have Vn −→ V , then it L−s

already holds (Vn , Yn ) −→ (V, Y ). L−s

(ii) Let d = 1 and Yn −→ Y = V U for some U ∼ N (0, 1) which is indeP

pendent of F . Further assume Vn −→ V with Vn , V > 0. Then it holds L−s

Yn /Vn −→ U . (iii) Assume that the random variable Y is G-measurable and that the G-stable √ L−s convergence n(Yn − Y ) −→ X holds. Then we also have √

L−s

n(g(Yn ) − g(Y )) −→ ∇g(Y )X. L−s

L−s

Proof. First we will prove part (i): By Lemma B.3 Yn −→ Y implies (V, Yn ) −→ P

(V, Y ). Further we obtain (Vn , Yn ) − (V, Yn ) −→ 0 and hence Lemma B.6 yields L−s

the joint convergence in law (Vn , Yn ) −→ (V, Y ). The claim (ii) directly follows from (i) using the continuous mapping theorem for stable convergence in law stated in Lemma B.5. For part (iii) observe that by the mean value theorem we can find ξn between Y and Yn with √

√ n(g(Yn ) − g(Y )) = ∇g(ξn ) n(Yn − Y ).

320

Appendix L−s

P

P

From Yn −→ Y we conclude |Yn − Y | −→ 0 which yields ξn −→ Y . Consequently P

by the continuity of g we also obtain g(ξn ) −→ g(Y ). Hence, by Lemma B.3 we √ L−s get (g(ξn ), n(Yn − Y )) −→ (g(Y ), X) and then Lemma B.5 yields the claim. The Delta-method for stable convergence in law stated in part (iii) of Proposition B.7 also illustrates the benefits of working with stable convergence in law instead of ordinary weak convergence. Usually, the Delta-method can only be applied if Y is constant. However, using stable convergence we obtain a result also for random F -measurable variables Y ; compare [40]. In the setting of high-frequency statistics the occurring limits are very often random; compare e.g. the results in Chapter 2. Notes. The above discussion of some elementary properties of stable convergence in law is mostly an adaption of Section 2.1 of [40]. The paper [40] gives an elementary and intuitive introduction to the concept of stable convergence in law and explains its role in the context of high-frequency statistics. A more formal discussion of the concept of stable convergence in law and its applications for proving convergences of stochastic processes can be found in [31] and [30].

C Triangular Arrays In this appendix chapter we state three results which are useful for proving convergence results for random variables which can be represented as sums of martingale differences. To formulate the results we need to introduce the notion of a triangular array. Definition C.1. A triangular array with accomodating filtrations is a double sequence of R-valued random variables (ζkn )k,n∈N for which filtrations (Gkn )k∈N0 , n ∈ N, exist such that ζkn is Gkn -measurable.  Let Nn , n ∈ N, be (Gkn )k∈N0 -stopping times. Then for triangular arrays with accomodating filtrations we can obtain simplified conditions for proving Nn 

P

ζkn −→ 0

k=1 n n based on the conditional expectations E[ζkn |Gk−1 ], E[(ζkn )2 |Gk−1 ]. In particular the following result holds which is part of Lemma 2.2.11 in [30].

Lemma C.2. Let (ζkn )k,n∈N be a triangular array with accomodating filtrations (Gkn )k∈N0 , n ∈ N, and let Nn , n ∈ N, be (Gkn )k∈N0 -stopping times. Then the two convergences Nn 

P

n E[ζkn |Gk−1 ] −→ 0,

k=1

Nn 

P

n E[(ζkn )2 |Gk−1 ] −→ 0,

k=1

imply Nn 

P

ζkn −→ 0.

k=1

Usually we consider some process (Xt )t≥0 which is adapted to a filtration (Ft )t≥0 . Further let (τk,n )k∈N0 , n ∈ N, be increasing sequences of stopping times. Then ζkn is some function of the behaviour of the process X in the (random) interval (τk−1,n , τk−1,n ] e.g. ζkn = f (Xτk,n − Xτk−1,n ) for some deterministic function f . By setting Nn = sup{k : τk,n ≤ T } we sum up the ζkn up to some time horizon T . Appropriate filtrations are then given by Gkn = Fτk,n . In this specific scenario the © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2019 O. Martin, High-Frequency Statistics with Asynchronous and Irregular Data, Mathematische Optimierung und Wirtschaftsmathematik | Mathematical Optimization and Economathematics, https://doi.org/10.1007/978-3-658-28418-3

322

Appendix

following result holds which even yields u.c.p. convergence, compare page 57 in [41], for the process which we obtain if we vary the time horizon T . This result is part of Lemma 2.2.12 in [30] or Lemma 4.2 in [28]. Lemma C.3. Let (τk,n )k∈N0 , n ∈ N, be increasing sequences of stopping times with τ0,n = 0. For t > 0 denote Nn (t) = sup{k : τk,n ≤ t} and let ζkn be Gτk,n -measurable random variables. Suppose (At )t≥0 is a process with continuous paths of finite variation and we have Nn (t)



u.c.p. E[ζkn |Gτk−1,n ] −→

Nn (t)

At ,

k=1

then it also holds



P

E[(ζkn )2 |Gτk−1,n ] −→ 0 ∀t ≥ 0,

k=1

Nn (t) k=1

u.c.p.

ζkn −→ At .

If the second moments do not vanish Nn asn in Lemmata C.2 and C.3 we obtain stable convergence in law of the sum i=1 ζk and the limit of the sum over the second conditional moments describes the asymptotic variance. The following result is taken from Theorem 2.6 in [40]; compare also Theorem 2.1 in [26]. Proposition C.4. Let (Ω, (Ft )t≥0 , P) be a filtered probability space and let ζkn be Ftk,n -measurable real-valued random variables where tk,n = k/n. We assume that the ζkn are in some sense ”fully generated”, compare page 6 of [40], by some (one-dimensional) Brownian motion W . Futher, suppose that there exist absolutely  of finite variation such that continuous processes F , G and a continuous process B the following conditions hold nt



P  E[ζkn |Ftk−1,n ] −→ B t,

(C.1)

k=1 nt



E[(ζkn )2 |Ftk−1,n ] − (E[ζkn |Ftk−1,n ])2



 u.c.p.

−→ Ft =

k=1 nt



E[ζkn (Wk/n

u.c.p.

− W(k−1)/n )|Ftk−1,n ] −→ Gt =

k=1 nt



u.c.p.

E[(ζkn )2 1{|ζkn |>ε} |Ftk−1,n ] −→ 0, ∀ε > 0,



t 0

t 0

(vs2 + ws2 )ds,

vs2 ds,

(C.2)

(C.3)

(C.4)

k=1 nt



k=1

u.c.p.

k/n − M (k−1)/n )|Ftk−1,n ] −→ 0 E[ζkn (M

(C.5)

C Triangular arrays

323

where (vs )s≥0 and (ws )s≥0 are predictable processes and the property (C.5) holds  with [W, M ] ≡ 0. Then the following F-stable for all (Ft )t≥0 -martingales M convergence holds Nn (t)



k=1

Ls ζkn −→

t + B





t 0

vs dWs +

t 0

s , t ≥ 0, ws dW

t )t≥0 denotes a standard Brownian motion defined on with Nn (t) = nt where (W ) which is independent of F .  F, P an extended probability space (Ω,