Advances in mathematical economics. Vol.20 [1st ed.] 9811004757, 978-981-10-0475-9, 978-981-10-0476-6, 9811004765

Table of contents :
Front Matter....Pages i-viii
Front Matter....Pages 1-1
Local Risk-Minimization for Barndorff-Nielsen and Shephard Models with Volatility Risk Premium....Pages 3-22
On a Fractional Differential Inclusion in Banach Space Under Weak Compactness Condition....Pages 23-75
On First-Order Partial Differential Equations: An Existence Theorem and Its Applications....Pages 77-87
Real Radicals and Finite Convergence of Polynomial Optimization Problems....Pages 89-99
Front Matter....Pages 101-101
On Differentiated and Indivisible Commodities: An Expository Re-framing of Mas-Colell’s 1975 Model....Pages 103-128
Front Matter....Pages 129-129
Survey of the Theory of Extremal Problems....Pages 131-150
Fourier Analysis of Periodic Weakly Stationary Processes: A Note on Slutsky’s Observation....Pages 151-180
Back Matter....Pages 181-189

Citation preview

Advances in Mathematical Economics  20

Shigeo Kusuoka Toru Maruyama  Editors

Advances in Mathematical Economics Volume 20

Managing Editors Shigeo Kusuoka

Toru Maruyama

The University of Tokyo Tokyo, JAPAN

Keio University Tokyo, JAPAN

Editors Robert Anderson University of California, Berkeley Berkeley, U.S.A.

Jean-Michel Grandmont CREST-CNRS Malakoff, FRANCE

Kunio Kawamata Keio University Tokyo, JAPAN

Charles Castaing Université Montpellier II Montpellier, FRANCE

Norimichi Hirano Yokohama National University Yokohama, JAPAN

Hiroshi Matano The University of Tokyo Tokyo, JAPAN

Francis H. Clarke Université de Lyon I Villeurbanne, FRANCE Egbert Dierker University of Vienna Vienna, AUSTRIA Darrell Duffie Stanford University Stanford, U.S.A. Lawrence C. Evans University of California, Berkeley Berkeley, U.S.A. Takao Fujimoto Fukuoka University Fukuoka, JAPAN

Tatsuro Ichiishi The Ohio State University Ohio, U.S.A. Alexander D. Ioffe Israel Institute of Technology Haifa, ISRAEL Seiichi Iwamoto Kyushu University Fukuoka, JAPAN Kazuya Kamiya The University of Tokyo Tokyo, JAPAN

Kazuo Nishimura Kyoto University Kyoto, JAPAN Yoichiro Takahashi The University of Tokyo Tokyo, JAPAN Akira Yamazaki Hitotsubashi University Tokyo, JAPAN Makoto Yano Kyoto University Kyoto, JAPAN

Aims and Scope. The project is to publish Advances in Mathematical Economics once a year under the auspices of the Research Center forMathematical Economics. It is designed to bring together those mathematicians who are seriously interested in obtaining new challenging stimuli from economic theories and those economists who are seeking effective mathematical tools for their research. The scope of Advances in Mathematical Economics includes, but is not limited to, the following fields: – Economic theories in various fields based on rigorous mathematical reasoning. – Mathematical methods (e.g., analysis, algebra, geometry, probability) motivated by economic theories. – Mathematical results of potential relevance to economic theory. – Historical study of mathematical economics. Authors are asked to develop their original results as fully as possible and also to give a clear-cut expository overview of the problem under discussion. Consequently, we will also invite articles which might be considered too long for publication in journals.

More information about this series at http://www.springer.com/series/4129

Shigeo Kusuoka • Toru Maruyama Editors

Advances in Mathematical Economics Volume 20

123

Editors Shigeo Kusuoka Professor Emeritus The University of Tokyo Tokyo, Japan

Toru Maruyama Professor Emeritus Keio University Tokyo, Japan

ISSN 1866-2226 ISSN 1866-2234 (electronic) Advances in Mathematical Economics ISBN 978-981-10-0475-9 ISBN 978-981-10-0476-6 (eBook) DOI 10.1007/978-981-10-0476-6

© Springer Science+Business Media Singapore 2016 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Science+Business Media Singapore Pte Ltd.

Preface

This volume of Advances in Mathematical Economics is basically a collection of selected papers presented at the 6th Conference on Mathematical Analysis in Economic Theory, which was held in Tokyo January 26–29, 2015. The conference was organized by the Japanese Society for Mathematical Economics. On behalf of the organization committee of the conference, we would like to extend our cordial gratitude to the Keio Economic Society, the Top Global University Project, and the Oak Society, Inc. for their generous financial support. And of course, it is a great pleasure for us to express our warmest thanks to all the participants of the conference for their contributions to our project. Professor C. Castaing’s paper and T. Maruyama’s paper were not read at the conference but are included in this issue. This volume contains the whole programme as well as some photographs taken on this occasion. December 14, 2015 S. Kusuoka, T. Maruyama Managing Editors of Advances in Mathematical Economics General Managers of the 6th Conference We formerly classified our international research meetings into two categories, “conference” and “workshop”. However, we relinquish this distinction from now on for the sake of simplicity and just use the term “conference”.

v

Contents

Part I

Research Articles

Local Risk-Minimization for Barndorff-Nielsen and Shephard Models with Volatility Risk Premium . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Takuji Arai

3

On a Fractional Differential Inclusion in Banach Space Under Weak Compactness Condition .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C. Castaing, C. Godet-Thobie, L.X. Truong, and F.Z. Mostefai

23

On First-Order Partial Differential Equations: An Existence Theorem and Its Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Yuhki Hosoya

77

Real Radicals and Finite Convergence of Polynomial Optimization Problems .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Yoshiyuki Sekiguchi

89

Part II

Expository Review

On Differentiated and Indivisible Commodities: An Expository Re-framing of Mas-Colell’s 1975 Model . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 103 M. Ali Khan and Takashi Suzuki Part III

Mini Courses

Survey of the Theory of Extremal Problems . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 131 V. Tikhomirov

vii

viii

Contents

Fourier Analysis of Periodic Weakly Stationary Processes: A Note on Slutsky’s Observation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 151 Toru Maruyama Programme .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 181 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 187

Part I

Research Articles

Adv. Math. Econ. 20, 3–22 (2016)

Local Risk-Minimization for Barndorff-Nielsen and Shephard Models with Volatility Risk Premium Takuji Arai

Abstract We derive representations of locally risk-minimizing strategies of call and put options for Barndorff-Nielsen and Shephard models: jump type stochastic volatility models whose squared volatility process is given by a non-Gaussian Ornstein-Uhlenbeck process. The general form of Barndorff-Nielsen and Shephard models includes two parameters: volatility risk premium ˇ and leverage effect . Arai and Suzuki (Local risk minimization for Barndorff-Nielsen and Shephard models. submitted. Available at http://arxiv.org/pdf/1503.08589v2) dealt with the same problem under constraint ˇ D  12 . In this paper, we relax the restriction on ˇ; and restrict  to 0 instead. We introduce a Malliavin calculus under the minimal martingale measure to solve the problem. Keywords Locally risk-minimizing strategy • Barndorff-Nielsen and Shephard models • Stochastic volatility models • Malliavin calculus • Lévy processes

JEL Classification: G11, G12 Mathematics Subject Classification (2010): 91G20, 60H07 T. Arai () Department of Economics, Keio University, Tokyo, Japan e-mail: [email protected] © Springer Science+Business Media Singapore 2016 S. Kusuoka, T. Maruyama (eds.), Advances in Mathematical Economics Volume 20, Advances in Mathematical Economics, DOI 10.1007/978-981-10-0476-6_1

3

4

T. Arai

Article Type: Research Article Received: June 4, 2015 Revised: December 12, 2015

1 Introduction Locally risk-minimizing (LRM, for short) strategies for Barndorff-Nielsen and Shephard models (BNS model, for short) are discussed. Here local riskminimization is a very well-known quadratic hedging method of contingent claims for incomplete financial markets. On the other hand, BNS models are stochastic volatility models suggested by Barndorff-Nielsen and Shephard [2, 3]. It is known that some stylized facts of financial time series are captured by BNS models. The square volatility process  2 of a BNS model is given as an Ornstein-Uhlenbeck process driven by a subordinator without drift, that is, a nondecreasing pure jump Lévy process. Thus,  2 is a jump process given as a solution to the following stochastic differential equation (SDE, for short): dt2 D t2 dt C dHt ; 02 > 0; where  > 0, H is a subordinator without drift. Now, we denote by S the underlying asset price process. The general form of S is given by Z St D S0 exp

t 0



  C ˇs2 ds C

Z

t 0

 s dWs C Ht ;

where S0 > 0, , ˇ 2 R,   0, W is a 1-dimensional Brownian motion. The last term Ht accounts for the leverage effect; and ˇs2 is called the volatility risk premium, which is considered as the compensation required by investors holding volatile assets. From the view of (2) below, the volatility risk premium vanishes when ˇ D  12 . So that, ˇ would take a value greater than or equal to  12 . For more details on BNS models, see Cont and Tankov [4] and Schoutens [11]. Our purpose is to obtain representations of LRM strategies of call and put options for BNS models under constraint  D 0 and no constraint on ˇ. On the other hand, Arai and Suzuki [1] studied the same problem under constraint ˇ D  12 and no constraint on . That is, they dealt with the case where volatility risk premium is not taken into account. To the contrary, we will treat BNS models with volatility risk premium. In other words, we relax the restriction on ˇ. Instead, we restrict  to 0, which induces the continuity of S. Then, S is written as Z St D S0 exp

t 0

   C ˇs2 ds C

Z

t 0

 s dWs :

(1)

Local Risk-Minimization for BNS Models

5

Actually, the continuity of S makes the problem easy to deal with. To calculate LRM strategies, we need to consider the minimal martingale measure (MMM, for short). When S is continuous, the subordinator H remains a Lévy process even under the MMM. On the other hand, the generalization of ˇ makes the problem complicated. When ˇ D  12 , the density process Z of the MMM is given as a solution to an SDE with the Lipschitz continuity. Thus, as shown in [1], Z has the Malliavin differentiability under P, which played a vital role in [1]. However, this property is not generalized to the case of ˇ ¤  12 . Hence, we need to take a different approach from [1]. In order to overcome this difficulty, making the best of the fact that the Lévy property of H is preserved, we innovate a Malliavin calculus under the MMM. As a result, we can calculate LRM strategies without attention to the property of Z. To our best knowledge, except for [1], there is only one preceding research on LRM strategies for BNS models: Wang, Qian and Wang [15]. Besides they treated the problem under the same parameter restrictions as ours, although they did not use Malliavin calculus. However, their discussion seems to be inaccurate mathematically. For example, they did not mention the SC condition defined in Sect. 2.1 of this paper, any condition on the Lévy measure of H, and any property of the MMM. These are indispensable to discuss LRM strategies. In addition, their expression of LRM strategies is given by a partial derivative of a function on t, St and t . It is very difficult to calculate it explicitly. On the other hand, we give a concrete representation for LRM strategies by using Malliavin derivative of put options. Outline of this paper is as follows. A precise model description and standing assumptions are given in Sect. 2. In Sects. 2.1, 2.2, and 2.3, we define an LRM strategy, the MMM and a Malliavin derivative, respectively. Our main results are provided in Sect. 3; and conclusions will be given in Sect. 4.

2 Preliminaries We consider a financial market model in which only one risky asset and one riskless asset are tradable. For simplicity, we assume that the interest rate is given by 0. Let T > 0 be the finite time horizon. The fluctuation of the risky asset is described as a process S given by (1). We consider a complete probability space .; F ; P/ with a filtration F D fFt gt2Œ0;T as the underlying space. Suppose that F is generated by Wt and Ht ; and satisfies the usual condition, that is, F is right continuous, and F0 contains all null sets of P. The asset price process S given in (1) is a solution to the following SDE: n o O 2 dt C t dWt ; dSt D St dt C ˇ t

(2)

h i Rt O 2 ds and Mt WD St  S0  At , where ˇO WD ˇ C 12 . Denoting At WD 0 Ss  C ˇ s we have St D S0 C Mt C At , which is the canonical decomposition of S. Further, we

6

T. Arai

denote Lt WD log.St =S0 / for t 2 Œ0; T, that is, Z t Z t 2 Lt D t C ˇ s ds C s dWs : 0

0

Defining Jt WDR Ht , we denote by N the Poisson random measure of J, that is, 1 we have Jt D 0 xN.Œ0; t; dx/. Denoting by  the Lévy measure of J, we have e that N.dt; dx/ WD N.dt; dx/  .dx/dt is the compensated Poisson random measure. Remark that N and  are defined on Œ0; T  .0; 1/ and .0; 1/, respectively; and .dx/ D  H .dx/, where  H is the Lévy measure of H. Moreover, Proposition 3.10 of [4] implies Z 1 .x ^ 1/.dx/ < 1: (3) 0

We need to impose the following standing assumptions on  as in [1]. As stated in Remark 2.2 below, the standing assumptions do not exclude representative examples of BNS models, although parameters are restricted. Assumption 2.1. (A1) The Lévy measure  is absolutely continuous with respect to the Lebesgue measure on .0; 1/. (A2) There exists a > 0 such that – > Œ2ˇO C C 1B.T/, – R  ˇO 2 B.T/, and 1 – 1 e2 x .dx/ < 1, Rt where B.t/ WD 0 es ds D

1et 

for t 2 Œ0; T.

Remark 2.2. 1. When ˇ D  12 , that is, ˇO D 0, (A2) is equivalent to the existence of " > 0 R1 such that 0 e.2C"/B.T/x .dx/ < 1. In [1] dealing with the case of ˇ D  12 , R 1 2B.T/x .dx/ < 1 is assumed in their Assumption 2.2, which is almost the 0 e same as the above (A2) for ˇ D  12 . 2. We do not need to assume conditions corresponding to the second condition of Assumption 2.2 in [1], which ensures the positivity of the density of the MMM defined below, since the continuity of S implies that the MMM becomes a probability measure automatically. 3. Condition (A2), together with Proposition 3.14 of [4], ensures EŒe2 JT  < 1, which obviously includes EŒ j2T  < 1. 4. Condition (A1) guarantees Assumption Z1 in Nicolato and Venardos [6], which we need in the proof of Lemma 2.9 below. 5. Assumption 2.1 does not exclude two representative examples of  2 , “IG-OU” and “Gamma-OU”. “IG-OU” is the case where  H is given as 3 1 2 a  H .dx/ D p x 2 .1 C b2 x/e 2 b x 1.0;1/ .x/dx; 2 2

Local Risk-Minimization for BNS Models

7

where a > 0 and b > 0. The invariant distribution of  2 follows an inverseGaussian distribution with a > 0 and b > 0. Then  2 is called an IG-OU process. If o n b2 > 2 Œ2ˇO C C 1 _ ˇO 2 B.T/; 2 then Assumption 2.1 is satisfied. Next, “Gamma-OU” is the case where the invariant distribution of  2 is given by a Gamma distribution with a > 0 and b > 0. In this case,  H is described as  H .dx/ D abebx 1.0;1/ .x/dx: As well as the IG-OU case, Assumption 2.1 is satisfied if o nh i b > 2 2ˇO C C 1 _ ˇO 2 B.T/: For more details on this topic, see also [6] and [11].

2.1 Local Risk-Minimization In this subsection, we define an LRM strategy. To this end, we define the SC condition firstly; and show that S satisfies it under Assumption 2.1. S is said to satisfy the SC condition, if the following three conditions hold:    1=2 R T  (a) ŒMT C 0 jdAs j 2 < 1. L .P/ R (b) A is absolutely continuous with respect to hMi, i.e. A D ƒdhMi for some process ƒ. Rt (c) The mean-variance trade-off process Kt WD 0 ƒ2s dhMis is finite, that is, KT is finite P-a.s. Proposition 2.3. S satisfies the SC condition under Assumption 2.1. Proof. First of all, we show item (a). We prepare the following lemma: RT Lemma 2.4. 0 t2 dt 2 Ln .P/ for any n  1. Proof of Lemma 2.4. Since we have Z

t 0

s2 ds D 02 D

Z

t

es ds C

0

02 B.t/

Z tZ 0

Z tZ

t

C

e 0

u

s 0

e.su/ dJu ds

.su/

dsdJu D

02 B.t/

 02 B.t/ C B.t/Jt  02 B.T/ C B.T/JT

Z

t

C 0

B.t  u/dJu (4)

8

T. Arai

for any t 2 Œ0; T, it suffices to show JT 2 Ln .P/ for any n  1. By Remark 2.2, we have EŒexpf2 JT g < 1, from which JT 2 Ln .P/ follows for any n  1.  Note that we have  Z  1=2 ŒM C  T "

T 0

2  jdAt j 2

L .P/

Z

 2E ŒMT C "Z  2E "  2E

T 0

sup

0sT

jdAt j

0

2 St t2 dt

Ss2

2 #

T

Z

T

C 0

(Z

T 0

t2 dt

St j C

O 2 jdt ˇ t

 Z O C jjT C jˇj

2 #

0

T

t2 dt

2 ) # :

If sup0sT Ss 2 L2a .P/ holds for a sufficiently small a > 1, item (a) holds by the Hölder inequality and Lemma 2.4. Now, we take an a > 1 such that (  ) C a2 2 2 aˇ C C a B.T/ < : (5) 2 Note that we can find such an a > 1 from the view of (A2) in Assumption 2.1. We shall see sup0sT Ss 2 L2a .P/. Equation (4) implies that  e

aLt

Z

t

D exp at C aˇ 0

s2 ds

Z

t

Ca 0

 s dWs

Z t   Z t Z a2 t 2 a2 2 D exp at   ds C a s dWs C aˇ C s ds 2 0 s 2 0 0  Z t Z a2 t 2 s ds C a s dWs  C exp  2 0 0  Z tZ 1 Z tZ 1 C bxe N.ds; dx/ C Œbx C 1  ebx .dx/ds 

0

DW

0

0

0

CYta;b ;

RT R1 2 C B.T/, and C WD expfajjT C b02 C 0 0 .ebx  1/ where b WD aˇ C a2 .dx/dtg. Taking into account (5) and (A2) in Assumption 2.1, Lemma 2.5 below

Local Risk-Minimization for BNS Models

9

yields that Y a;b is a square integrable martingale. Thus, Doob’s inequality yields " E

# sup

0sT

Ss2a

" DE

"

# S02a

sup e

2aLs



0sT

S02a C2 E

# sup

0sT

. Ysa;b /2

 4S02a C2 EŒ. YTa;b /2  < 1: O t2 1 Cˇ , St t2

Next, taking ƒt WD Noting that

t2

we have item (b). Moreover, we see item (c).

is represented as t2 D et 02 C

Z 0

t

e.ts/ dJs ;

we have that t2  et 02  eT 02 for any t 2 Œ0; T. Thus, together with Lemma 2.4, item (c) follows.

(6) 

Lemma 2.5. For a 2 R and b  0, we denote 

Z t Z tZ 1 Z a2 t 2 s ds C a s dWs C bxe N.ds; dx/ 2 0 0 0 0  Z tZ 1 bx C Œbx C 1  e .dx/ds :

Yta;b WD exp



0

0

1. If a and b satisfy Z

1 1

   a2 exp 2b C B.T/ x .dx/ < 1; 2

(7)

then the process Y a;b is a martingale. 2. When we strengthen (7) to Z

1 1

expf.4b C 2a2 B.T//xg.dx/ < 1;

Y a;b is a square integrable martingale.

(8)

10

T. Arai

Proof. 1. Theorem 1.4 of Ishikawa [5] introduced a sufficient condition for Y a;b to be a martingale, which condition consists of the following three items: R1 (a) R0 Œb2 x2 C .1  ebx /2 .dx/ < 1, 1 bx (b) 0h Œebx n bx C 1  eoi .dx/ < 1, and R 2 T a 2 (c) E exp 2 0 t dt < 1. By (3) and (7), (b) are satisfied. Next, (7) and Proposition 3.14 in h items n 2 (a) andoi [4] imply E exp a2 B.T/JT < 1, from which item (c) follows by (4). 2. We have only to show the square integrability of Y a;b . To this end, we firstly obtain an upper estimate of .YTa;b /2 , and confirm its integrability. Denoting  WD 2b C a2 B.T/, we have . YTa;b /2 D exp



 a2

Z

T

Z

T 0

Z

1

 exp

0

0

 2a2

Z

T

C  D exp

Z

0

Z

 2a2

Z

T

Z

0

T 0



0

Z

T

D exp 0

Z 0

T

t dWt

0

s2 ds C 2a

Z

1

s2 ds C 2a

 x C 2  2e

Z

0

Z



1

2Œbx C 1  ebx .dx/dt

0

t dWt C a2 02 B.T/ C a2 B.T/JT

Z

Z

T

Z



1

2Œbx C 1  e .dx/dt bx

0

0

T

0 bx

T

0 T

2bxe N.dx; dt/ C

Z

1

C 0

T

1 0

Z

2bxe N.dx; dt/ C

C 

s2 ds C 2a



t dWt C .dx/dt C

Z

T 0

Z

1 0

 xe N.dx; dt/

a2 02 B.T/



  2a; bx x 2 2 1  2e C e .dx/dt C a 0 B.T/ YT :

R1 Under (8), we have 1 expf2 xg.dx/ < 1. Thus, we can see that Y 2a; is by the same sort argument as item 1. Moreover, we have R 1a  martingale 1  2ebx C e x .dx/ < 1, from which the square integrability of YTa;b 0 follows.  Next, we give a definition of an LRM strategy based on Theorem 1.6 of Schweizer [13].

Local Risk-Minimization for BNS Models

11

Definition 2.6. 1. ‚Sh denotes the space of alli R-valued predictable processes satisfying RT RT E 0 t2 dhMit C . 0 j t dAt j/2 < 1.

2. An L2 -strategy is given by a pair ' D . ; /, where 2 ‚S and is an adapted process such that V.'/ WD S C is a right continuous process with EŒVt2 .'/ < 1 for every t 2 Œ0; T. Note that t (resp. t ) represents the amount of units of the risky asset (resp. the risk-free asset) an investor holds at time t. 3. For claim F 2 L2 .P/, the process CF .'/ defined by CtF .'/ WD F1ftDTg C Vt .'/  Rt 0 s dSs is called the cost process of ' D . ; / for F. 4. An L2 -strategy ' is said a locally risk-minimizing strategy for claim F if VT .'/ D 0 and CF .'/ is a martingale orthogonal to M, that is, ŒCF .'/; M is a uniformly integrable martingale. 5. An F 2 L2 .P/ admits a Föllmer-Schweizer decomposition (FS decomposition, for short) if it can be described by Z F D F0 C

T 0

tF dSt C LFT ;

(9)

where F0 2 R, F 2 ‚S and LF is a square-integrable martingale orthogonal to M with LF0 D 0. For more details on LRM strategies, see Schweizer [12, 13]. Now, we introduce a relationship between an LRM strategy and an FS decomposition. Proposition 2.7. Under Assumption 2.1, an LRM strategy ' D . ; / for F exists if and only if F admits an FS decomposition; and its relationship is given by Z t D

tF ;

t D F0 C

t 0

sF dSs C LFt  F1ftDTg  tF St :

Proof. This is from Proposition 5.2 of [13], together with Proposition 2.3.



Thus, it suffices to get a representation of in (9) in order to obtain an LRM strategy for claim F. Henceforth, we identify F with an LRM strategy for F. F

2.2 Minimal Martingale Measure We need to study the MMM in order to discuss FS decomposition. A probability measure P  P is called the minimal martingale measure (MMM), if S is a P -martingale; and any square-integrable P-martingale orthogonal to M remains a martingale under P . Now, we consider the following SDE: dZt D Zt ƒt dMt ; Z0 D 1:

(10)

12

T. Arai

The solution to (10) is a stochastic exponential of  denoting ut WD ƒt St t D

R 0

ƒt dMt . More precisely,

 O t C ˇ t

(11)

for t 2 Œ0; T, we have ƒt dMt D ut dWt ; and  Zt D exp

Z

1 2



t 0

u2s ds 

Z

 us dWs :

t 0

(12)

To see that ZT becomes the density of the MMM, it is enough to show that Z is a square integrable martingale. Proposition 2.8. Z is a square integrable martingale. Proof. It is enough to see the square integrability of ZT , and confirm if Z satisfies Novikov’s criterion from the view of Theorem III.41 in Protter [9]. First, we see ZT 2 L2 .P/. By (6), there is a constant Cu > 0 such that 2 C 2ˇO C ˇO 2 t2  Cu C ˇO 2 t2 t2

u2t D by (11). Thus, (12) implies  Z ZT2 D exp 2 

Z

T 0 T

 exp 2 0



Z

T

 exp 2 0



u2t dt  u2t dt  u2t dt 

Z Z

Z

T

2ut dWt C

0 T

0

Z

0

T

 exp Z

T

Z 2

Z

1

C 0

0

T 0

0

u2t dt



2ut dWt C TCu C ˇO 2

Z

T 0

t2 dt



 2ut dWt C TCu C ˇO 2 Œ02 B.T/ C B.T/JT 

 exp TCu C ˇO 2 02 B.T/ C 

T

u2t dt 

Z

T 0

Z

Z

1 0

Œe x  1 .dx/dt

T 0

2ut dWt C

 Πx C 1  e x  .dx/dt ;

Z 0

T

Z

1 0



xe N.dx; dt/

Local Risk-Minimization for BNS Models

13

since ˇO 2 B.T/  by (A2). In addition, Remark 2.2 implies

 Z E exp 2

0

T

u2t dt



 Z  E exp 2TCu C 2ˇO 2

T 0

t2 dt



˚    exp 2TCu C 2 02 E e2 JT < 1:

(13)

Hence, we can see that ZT2 is integrable by the same manner as the proof of item 1 in Lemma 2.5. Lastly, remark that (13) gives Novikov’s criterion. Thus, Proposition 2.8 follows.  

Henceforth, we denote the MMM by P , that is, we have ZT D dP dP . Note that  N remains a martingale dWtP WD dWt C ut dt is a Brownian motion under P ; and e  under P . Remark that we can rewrite (2) and LT as dSt D St t dWtP and LT D R RT  1 T 2 P 0 s dWs  2 0 s ds, respectively. The following two lemmas are indispensable to formulate a Malliavin calculus under P . RtR1   N; and WtP C 0 0 ze N.ds; dz/.DW Xt / is a Lemma 2.9. W P is independent of e  Lévy process under P . Proof. This is given from Theorem 3.2 in [6]. Remark that Assumptions Z1 – Z3 in [6] are their standing assumptions. Assumptions Z1 and Z2 are satisfied in our setting from Assumption 2.1. On the other hand, Assumption Z3 does not necessarily hold, but it is not needed for Theorem 3.2 in [6].  Lemma 2.10. Let F be the augmented filtration generated by X  . Then, the filtration F coincides with F . 

Proof. Since W P is F-adapted, it is clear that F  F . Next, we have 

Wt D WtP 

Z

t 0



us ds D WtP 

Z t 0

  O s ds C ˇ s

by (11). W is then F -adapted, since  is F -adapted. Thus, we have F F .



2.3 Malliavin Calculus Under P Here, regarding .; F ; P / as the underlying probability space, we formulate a Malliavin calculus for X  under P based on Petrou [8] and Chapter 5 of Renaud [10] from the view of Lemmas 2.9 and 2.10. Although [1] adopted the canonical Lévy space framework undertaken by Solé et al. [14], we need to take a different way to define a Malliavin derivative, since the property of the canonical Lévy space is not preserved under change of measure.

14

T. Arai

First of all, we need to prepare some notation; and define iterated integrals with  respect to W P and e N. Denoting U0 WD Œ0; T and U1 WD Œ0; T  .0; 1/, we define Z Q0 .A/ WD A

Z

dWtP



for any A 2 B.U0 /;

e N.dt; dx/

Q1 .A/ WD

for any A 2 B.U1 /;

A

hQ0 i WD m;

hQ1 i WD m  ;

and

where m is the Lebesgue measure on U0 . We denote ( G. j1 ;:::;jn / WD

j .u11 ; : : : ; unjn /

2

n Y

) Ujk W 0 < t1 <    < tn < T

kD1 j

for n 2 N and . j1 ; : : : ; jn / 2 f0; 1gn , where ukk WD tk if jk D 0; and WD .tk ; x/ if jk D 1 for k D 1; : : : ; n. We define an n-fold iterated integral as follows: Jn. j1 ;:::;jn / .g.nj1 ;:::;jn / /

Z

g.nj1 ;:::;jn / .u11 ; : : : ; unjn /Qj1 .du11 /    Qjn .dunjn /; j

WD G. j1 ;:::;jn /

j

  N . j ;:::;j / where gn 1 n is a deterministic function in L2 G. j1 ;:::;jn / ; nkD1 hQjk i . Then, Theorem 1 in [8] ensures that every L2 .P / random variable F is represented as . j ;:::;j / a sum of iterated integrals, that is, we can find deterministic functions gn 1 n 2   N n L2 G. j1 ;:::;jn / ; kD1 hQjk i for n 2 N and . j1 ; : : : ; jn / 2 f0; 1gn such that F has the following chaos expansion: F D EP ŒF C

1 X

X

nD1 . j1 ;:::;jn

Jn. j1 ;:::;jn / .g.nj1 ;:::;jn / /:

(14)

/2f0;1gn

Note that the infinite series in (14) converges in L2 .P /. Now, we define D0 the space of Malliavin differentiable random variables; and a Malliavin derivative operator D0 . Denoting, for 1  k  n and t 2 .0; T/, ˚ j jkC1 jk1 Gk. j1 ;:::;jn / .t/ WD .u11 ; : : : ; uk1 ; ukC1 : : : ; unjn / 2 G. j1 ;:::;jk1 ;jkC1 ;:::;jn / W  0 < t1 <    < tk1 < t < tkC1 <    < tn < T ;

Local Risk-Minimization for BNS Models

15

we define D0 as  1 X 0 D WD F 2 L2 .P /; F D EP ŒF C

X

Jn. j1 ;:::;jn / .g.nj1 ;:::;jn / / W

nD1 . j1 ;:::;jn /2f0;1gn .0/

kg1 k2L2 .m/ C

1 X

X

n X

1f jk D0g

nD2 . j1 ;:::;jn /2f0;1gn kD1

Z

T

 0

 2  . j1 ;:::;jk1 ;0;jkC1 ;:::;jn /  .: : : ; t; : : : / 2 gn L

Gk. j

1 ;:::;jn /

.t/

dt

 0, we have .K  ST /C 2 D0 , and D0t .K  ST /C D 1fST log.K=S0 /:

we have that fK 2 C1 .R/ and 0 < fK0 .r/  K for any r 2 R. Thus, Theorem 2 of [8] implies that fK .LT / 2 D0 and D0t fK .LT / D fK0 .LT /D0t LT D fK0 .LT /t :

(15)

Since .K  ST /C D .K  fK .LT //C , we need only to see .K  fK .LT //C 2 D0 ; and calculate D0t .K  fK .LT //C . To this end, we take a mollifier Rfunction ' which 1 is a C1 -function from R to Œ0; 1/ with supp.'/ Œ1; 1 and 1 '.x/dx D 1. R1 C We denote 'n .x/ WD n'.nx/ and gn .x/ WD 1 .K  y/ 'n .x  y/dy for any n  1. Noting that Z gn .x/ D

Z 1 y C y

KxC K xC '. y/dy; '. y/dy D n n 1 n.Kx/ 1

R1 we have g0n .x/ D  n.Kx/ '. y/dy, so that gn 2 C1 and jg0n j  1. Thus, Theorem 2 in [8] again implies that, for any n  1, gn . fK .LT // 2 D0 and D0t gn . fK .LT // D g0n . fK .LT //D0t fK .LT / D g0n . fK .LT //fK0 .LT /t by (15). We have then sup kD0 gn . fK .LT //k2L2 .mP /  K 2 EP n1

Z

T 0

 t2 dt < 1:

(16)

Local Risk-Minimization for BNS Models

17

In addition, noting that ˇ   ˇ y C KxC  .K  x/C '. y/dyˇˇ n 1 Z 1 1 1 j yj'. y/dy   n 1 n

ˇZ ˇ jgn .x/  .K  x/C j D ˇˇ

1

for any x 2 R, we have limn!1 EŒjgn . fK .LT //  .K  fK .LT //C j2  D 0. As a result, Lemma 3.2 below implies that .K  fK .LT //C 2 D0 . Furthermore, Lemma 2 of [8] ensures the existence of a subsequence nk such that D0 gnk . fK .LT // converges to D0 .K  fK .LT //C in the sense ofR L2 .m  P /. On the other hand, we have 1 limn!1 g0n .x/ D 1fx 0/. Then there exists a neighborhood Lxx .Ox; /Œx; W Y of 0 2 Y and mapping y 7! .x.y/; .y//, defined on W, such that O Lx .x.y/; .y// D 0, x.0/ D xO ; .0/ D uF.x.y// D y. The family fx.y/g is called field of extremals. Proof. Consider mapping ˆ.x; y/ D .Lx .Ox C x;b

C /; F.Ox C x//. It maps a neighborhood e V of .0; 0/ 2 X  Y  D X  Y into X   Y D H. O C We have: ˆ.0; 0/ D .0; 0/; ˆ 2 C1 and ˆ0 .0; 0/Œx;  D .Lxx .Ox; /x 0  0 0 . f .Ox// ; F .Ox/x/. It is possible to prove that ˆ .0; 0/ is a homeomorfic mapping of

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XY on itself and to put ˆ1 .0; y/ D .x.y/; .y// for which Lx .Ox Cx.y/;b

C .y// D 0; F.Ox C x.y// D y. t u Put S.y/ D f .x.y//. After differentiating we obtain that S0 .y/ D  .y/.

3.2 Conditions of Extrema for the Simplest Problem of Calculus of Variations Consider the following problem: Zt1 J.x.// D

L.t; x.t/; xP .t//dt ! min; x.t0 / D x0 ; x.t1 / D x1 ;

(P1 )

t0

where integrand L W RRR and a function xO ./, which is suspected to be minimum in the problem (P1 ) (in the space C1 .Œt0 ; t1 /, are smooth enough. Let y./ 2 C01 .Œt0 ; t1 / D f y./ 2 C1 .Œt0 ; t1 / j y.t0 / D y.t1 / D 0g. After Rt1 2 repeated differentiation we obtain K.y.// D d L.Ox./C#y.// j D0 D .A.t/Py.t/2 C d 2 t0

B.t/y2 .t//dt  0 (where A.t/ D b LxP xP .t/; B.t/ D b Lxx  dtd b LxPx .t/ are continuous). In other words function yO ./ 0 is a solution of the problem Zt1

.A.t/Py2 .t/ C B.t/y2 .t//dt ! min y./ 2 C01 .Œt0 ; t1 /:

(PQ1 )

t0

Application of Lagrange’s principle (, Pontryagin maximum principle) to this problem leads to Euler equation 

d .A.t/Py/ C B.t/y D 0; y.t0 / D y.t1 / D 0 dt

(it is called Jacobi equation for problem (PQ1 )) and Legendre condition A.t/  0 [28]. We permit further that strong Lagrange condition A.t/ > 0 8t 2 Œt0 ; t1  is satisfied. There exists the unique solution y./ of Jacobi equation with boundary conditions y.t0 / D 0; yP .t0 / D 1. It is very easy to reduce from Pontryagin maximum principle, that if xO ./ locally minimize problem .P1 /, then y.t/  0 8t 2 .t0 ; t1 /. This fact is called Jacobi necessary condition of weak minimum in problem .P1 /[18].

Survey of the Theory of Extremal Problems

147

Now we suppose that strong Jacobi condition (y.t/ > 00 8t 2 .t0 ; t1 /) is satisfied. In this case the quadratic functional K is represented in the following form: Zt1 K. y.// D

A.t/. yP .t/ C

P y.t/ y.t//2 dt; y.t/

t0

thus it affords the strong minimum for (PQ1 ) and from the general theorem on field extremals it follows that fx.t; y/ D y.ty1 / y.t/g is the field of extremals which covers all strip .t0 ; t1   R. Closeness of integrants of functionals J and K leads to field of extremals of the problem (PQ1 ) which covers a neighborhood of the extremal xO ./. Formula for derivative of S-function if it apply to S-function of the simplest problem of CA we obtain to Hamilton–Jacobi equation and then to Weierstrass formula for increment of functional of the problem (PQ1 ). It leads to the following result of Weierstrass [42]: Theorem 3.2. Strong conditions of Lagrange and Jacobi together with quasi regularity of integrand L are sufficient conditions of strong local minimum of extremal xO ./.

4 Existence (Principles of Compactness, Contractibility, Topology, Order and Monotonicity) I am sure that it would be possible to realize proofs of existence of solutions by means of some general principle of existence (similar to Dirichlet principle). And may be it will approach us to answer the following question: whether every regular problem has a solution (may be in some extended sense) D. Hilbert [16]

Main thesis: Hilbert’s «general principle of exitence» is in my opinion Weierstrass–Baire principle of compactness, according to which a lower semicontinuous function on a compact is bounded from below and attains its minimum. We illustrate application of this principle explaining existence theorem of solution of the simplest problem of calculus of variations: Zt1 L.t; x.t/; xP .t//dt ! min; x.ti / D xi ; i D 0; 1:

J.x.// D

(P1 )

t0

Theorem 3 (on existence in calculus of variations). If Lagrangian L in (P1 ) satisfies conditions of quasi regularity (this means that functions y 7! L.t; x; y/ are

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convex) and growth (i.e. L.t; x; y/  ak ykp Cb; a > 0; p > 1), then the problem (P1 ) has absolutely continuous solution. In this result growth provides compactness and regularity provides semicontinuity. But really almost all results mentioned above are nothing else but theorems on existence (of inverse mappings, of a hyperplane which separates convex bodies, Lagrange multipliers, fields of extremals or solutions of problems). These «existences» are based on the following principles of existence: 1. Weierstrass–Baire compacness principle. This principle was formulated above. 2. Brouwer topological principle. Let F 2 C.Bn ; Rn /; Bn D fx 2 Rn j jxj  1g such that jx  F.x/j < 1  ı; 0 < ı < 1: Then for all y 2 Rn ; j yj < ı there exists xO such that y D F.x/. 3. Baire ordered principle. Let .X; d/ be a complete metric space and let fAk gk2N be a family of nowhere dense sets in X. Then there exists a point xO 2 X n [k2N Ak : 4. Minty–Brouder monotonicity principle. A strictly monotone, continuous and coercitive mapping F W Rn ! Rn has a unique solution of the equation F.x/ D y for each y 2 Rn (and it can be reached by means of some special iterative procedures) 5. Banach contraction mapping principle. Complete metric space with a contraction mapping admits a unique fixed-point. And now I want to give a short trip trough proofs of our main results. Proofs of right-inverse mapping and generalized implicit function theorems are builded on combination of lemma on right-inverse mapping in linear case (which in turn is settled on Baire ordered principle) and modify Newton method (which is based on contractibility). Existence of Lagrange multipliers and construction of fields of extremals are based on theorems on inverse mappings. Existence of different objects in convex analysis is settled on monotonicity. Existence of solutions are relying on compactness. Convexity of images of optimal control mappings can be proved by combining of «almost convexity» of integral mappings and correction them with the help of Brouwer principle.

5 Algorithms (Methods of Solutions) Main thesis. There are two the most important approaches to numerical solution of extremal problems. One is called Direct methods. It is based on approximation of infinite dimensional problems by finite dimensional ones. The second approach is connected with solution of equations which follows from necessary conditions. In direct methods there are three the most essential ideas of reaching the goal: the idea of reasonable decent, ideas of using monotonicity and ideas of penalization.

Survey of the Theory of Extremal Problems

149

Among the most well-known methods there are simplex method of Dantzig [9] for solution of linear programming problems (this method realizes an idea of reasonable descent), Levin–Newman section method [30, 37], ellppsoid method of Shor–Yudin–Nemirovsky (they use ideas of monotonicity), interior method of Karmarkar of convex optimization based on idea of penalization. Application of the theory of extremal problems Let us turn back to the beginning, where were formulated reasons which stimulated persons to solve and investigate extremal problems. Now I want to formulate talking points about this. • Laws of dynamical processes (in optic, mechanics, hydro and aerodynamics an so on) are described by variational principles. So the mathematical theory of «naturphilosophy» (using the word of Newton) are expressed on the language of calculus of variations. • Rules of control over dynamical processes (such as cosmic navigation) are given by variational principles with constraints which characterize limitations human possibilities, so this processes often can be formalized as problems of optimal control. • Linear programming and optimal control became a mathematical basis of mathematical economics. • The majority of problems which were solved without computers have a standard solution basing on principles of the theory of extremum.

References 1. Euclid (2007) Euclid’s elements. Green Lion Press, Ann Arbor 2. Alexeev V, Tikhomirov V, Fomin S (1987) Optimal control. Plenum Publishing Corporation, New York 3. Banach S (1932) “Théorie des opérations linéaire”, Warszava, Monographje Matematyczne 4. Bernoulli I (1696) Problema novum, ad cujus solutionem Matematici invitantur. Acta Eruditorum 5. Bliss GA (1963) Lectures on the calculus of variations. University of Chicago Press 6. de Fermat P (1891) Oeuvres de Fermat, vol 1. Gauthier-Villars, Paris 7. Dini U (1877/1878) Analisi infinitesimale. Lezzione dettate nella Università Pisa. Bd 2 8. Dmitruk AV, Milyutin AA, Osmolovskii NP (1980) Lyustrenik’s theorem and the theory of extrema. Uspehi Mat Nauk 35(6): 11–46 9. Dantzig GB (1963) Linear Programming and Extensions. Princeton University Press, Princeton, NJ 10. Dubovitskii AY, Milyutin AA (1965) Extremum problems with constraints. Zh Vychisl Mat i Mat Fiz 5:395–453 11. Euler L (1744) Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latssimo sencu accepti. Lausanne 12. Fenchel W (1953) Convex Cones, Sets, and Functions. Princeton University Department of Mathematics, Princeton, NJ 13. Frèchet V (1912) Sur la notion de differentielle. Nouvelle annale de mathematique, Ser. 4, V.XII. S. 845

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14. Graves LM (1950) Some mapping theorems. Duke Math J 17:111–114 15. Hamilton WR (1835) Second essay on a general methods in dynamics. Philos Trans R Soc Pt 1:95–144 16. Hilbert D. Hazewinkel M (ed) (2001) “Hilbert problems”, Encyclopedia of mathematics. Springer. ISBN:978-1-55608-010-4 17. Ioffe A, Tikhomirov V (1979) Theory of extremal problems. North-Holland, Amsterdam 18. Jacobi CGJ (1837) Zur Theorie der Variations-Rechnung und der Differential-Gleichungen. Krelle’s Journall 17:68–82 19. John F Extremal problems with inequalities as subsidery conditions. In: Studies and Essays. Courant Anniverrsary Volume, 1948, pp 187–204 20. Kantorovich LV (1939) Mathematical methods of organizing and planning production. Manag Sci 6(4)(Jul., 1960):366–422 21. Karush WE (1939) Minima of functions of several variables with inequalities as side conditions, University of Chicago Press 22. Kepler I (1615) The volume of a Wine Barrel – Kepler’s Nova stereometria doliorum vinariorum, Lincii (Roberto Cardil Matematicas Visuales) 23. Kneser A (1925) Lehrbuch der Variationsrechnung. Springer, Wiesbaden 24. Kuhn HW, Tucker AW (1951) Nonlinear programming. University of California Press, Berkley, pp 481–482 25. Lagrange JL (1766) Essai d’une nouvelle méthode pour determiner les maxima et les minima periales Petropolitanae, vol 10, 51–93 26. Lagrange JL (1797) Théorie des fonctions analytiques, Paris 27. Leach E (1961) A note on inverse mapping theorem. Proc AMS 12:694–697 28. Legendre AM (1786) Mémoire sur la maniere de distingue les maxima des minima dans le calcul de variations/ Histoire de l’Academie Royallle des Sciences. Paris, pub 1788. 7–37 29. Leibniz G (1684) Acta Eroditorum, L.M.S., T. V, pp 220–226 30. Levin AY On an algorithm for the minimization of convex function Sov. Math., Dokl. – 1965. – no. 6, pp 268–290 31. Lyapunov AA (1940) O vpolnye additivnykh vyektor-funktsiyakh: // Izvyestiya Akadyemii nauk SSSR. Syer. matyematichyeskaya. 4(6):465–478 32. Lyusternik LA (1934) On constrained exstrema of functionals (in Rusian). Matem. Sbornik 41(3):390–401. See also Russian Math Surv 35(6):11–51 (1980) 33. Mayer A (1886) Begründung der Lagrangesche Multiplikatorenmethode der Variatinsrehbung. Math Ann 26 34. Minkovski H (1910) Geometrie der Zahlen. Teubner, Leipzig 35. Monge G (1781) Memoire sur la théorie des déblais et des remblais, Paris 36. Moreau JJ (1964) Fonctionelles sus-differènciables. C R Acad Sci (Paris) 258:1128–1931 37. Newman DJ (1965) Location of the maximum on unimodal surfices. J ACM 12 No 3:395–398 38. Newton I (1999) Mathematical principles of natural philosophy. University of California Press, Berkeley 39. Newton I, Whiteside DT (1967–1982). The mathematical papers of Isaac Newton, 8 vols. Cambridge University Press, Cambridge. ISBN:0-521-07740-0 40. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) The mathematical theory of optimal processes (Russian). English translation: Interscience 41. Rockafellar RT (1997) Convex analysis. Princeton landmarks in mathematics. Princeton University Press, Princeton 42. Weierstrass K (1927) Mathematische Werke, Bd 7. Vorlesungemn uber Variationsrehtung. Akad. Verlag, Berlin–Leipzig 43. Avakov ER, Magaril-Il’yaev GG, Tikhomirov VM (2013) Lagrange’s principle in extremum problems with constraints. Usp Mat Nauk 68(3):5–38

Adv. Math. Econ. 20, 151–180 (2016)

Fourier Analysis of Periodic Weakly Stationary Processes: A Note on Slutsky’s Observation Toru Maruyama

Abstract The periodic behavior of a specific weakly stationary stochastic process (w.s.p.) is examined from a viewpoint of classical Fourier analysis. (1) A w.s.p. has a spectral measure which is absolutely continuous with respect to the Lebesgue measure if and only if it is a moving average of a white noise. (2) A periodic or almost periodic w.s.p. must have a “discrete” spectral measure. Combining these two, we can conclude that any moving average of a white noise can neither be periodic nor almost periodic. However any w.s.p. can be approximated by a sequence of almost periodic w.s.p.’s in some specific sense. Keywords Weakly stationary process • Periodicity • Almost periodicity • Spectral measure

Article Type: Mini Course Received: October 8, 2015 Revised: October 28, 2015

T. Maruyama () Keio University, Tokyo, Japan e-mail: [email protected] © Springer Science+Business Media Singapore 2016 S. Kusuoka, T. Maruyama (eds.), Advances in Mathematical Economics Volume 20, Advances in Mathematical Economics, DOI 10.1007/978-981-10-0476-6_7

151

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T. Maruyama

Introduction During a decade around 1930, the world economy was thrown into the serious depression, nobody had ever experienced. A number of economic theories were proposed in order to analyze and control the violence of business fluctuations. Among them, the work of Eugen Slutsky [30], an Ukrainian mathematician, is particularly remarkable from the mathematical point of view.1 It was published in Econometrica, 1937. He tried to explain more or less regular fluctuations of macro-economic movements based upon the overlapping effects of random shocks. However, frankly speaking, his analysis was rather experimental and devoid of mathematical rigor. The purpose of the present paper is to give a systematic exposition of the mathematical skeleton of the Slutsky problem being based upon classical Fourier analysis. Our main conclusions are two-fold. (1) Any weakly stationary stochastic process generated by moving average of white noise (call Slutsky process) can neither be periodic nor almost periodic. If a weakly stationary process is periodic or almost periodic, its spectral measure must be discrete (i.e. a weighted sum of Dirac measures). However the spectral measure of a Slutsky process must be absolutely continuous with respect to the Lebesgue measure and hence it has a spectral density function. It immediately follows that any Slutsky process can never be periodic and even the almost periodicity is impossible. The Bochner-Herglotz theory concerning Fourier representation of positive semi-definite functions provide us the key analytical tool. (2) Slutsky’s conclusion is correct in a sophisticated sense. That is, the spectral measure of any weakly stationary process can be approximated by a sequence of spectral measures of almost periodic processes.

The earlier draft of this paper was read at the 20th Conference of the International Federation of Operational Research Societies, which was held in Barcelona, (July 13–18, 2014). It is a pleasure for me to express my cordial gratitude to the late Professor Tatsuo Kawata for incessant encouragement, which led me to the field of Fourier analysis. I am much indebted to Professor Shigeo Kusuoka for his kind suggestions concerning probability theory, which I am not familiar with very well. Helpful comments by Dr. Yuhki Hosoya and Mr.Chaowen Yu are also gratefully acknowledged. JEL Classification: CO2, E32 Mathematics Subject Classification (2010): 42A38, 42A82, 60G10 1

Frisch [11] also deserves a special attention.

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153

These positive claims are developed in the last two sections. I am much indebted to several forerunners in this discipline, in particular to the late Professor Tatsuo Kawata for basic ideas. I have to confess, however, that there still remain complicated reasonings which do not seem convincing enough for me. I try to reorganize these puzzling aspects as coherently as possible. The first six sections cover the fundamental concepts and results which are not so familiar to mathematical economists but are indispensable prerequisites for understanding last two sections.2

1 Basic Concepts Let (˝; E ; P) be a probability space, and T a subset of R. T is usually interpreted as the space of time and, in this paper, T is assumed to be either R or Z. A function X W T  ˝ ! C is said to be a stochastic process if the function ! 7! X.t; !/ is .E ; B.C//-measurable for any fixed t 2 T, where B.C/ is the Borel -field in C. C; R and Z denote the sets of complex numbers, real numbers and integers, respectively. The expectation EX.t; !/ of the stochastic process X.t; !/ at t 2 T is defined by Z EX.t; !/ D

˝

X.t; !/dP:

The function  W T  T ! C defined by .s; t/ D EŒX.s; !/  EX.s; !/ŒX.t; !/  EX.t; !/;

s; t 2 T

is called the covariance function of X.t; !/. Let T be the set of all the finite tuples of elements of T, that is T D ft D .t1 ; t2 ;    ; tn / j tj 2 T;

j D 1; 2;    ; n;

n 2 Ng:

Xt .!/ denotes the vector Xt .!/ D .X.t1 ; !/; X.t2 ; !/;    ; X.tn ; !//;

t2T:

The set function Xt W B.Cn / ! R defined by Xt .E/ D Pf! 2 ˝jXt .!/ 2 Eg;

E 2 B.Cn /

2 Kawata [17, 18], Maruyama [23] and Wold [31] are classical works on Fourier analysis of stationary stochastic processes, which provided me with all the basic mathematical background. Among more recent literatures, I wish to mention Brémaud [5]. Granger and Newbold [12] Chap. 2, Hamilton [13] Chap. 3 and Sargent [26] Chap. XI are textbooks written from the standpoint of economics.

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is a measure on B.Cn /, called the distribution of Xt .!/. The set of all the distributions fXt jt 2 T g is called the system of finite dimensional distributions determined by X.t; !/. A stochastic process X.t; !/ is strongly stationary if the distribution of .X.t1 C t; !/; X.t2 C t; !/;    ; X.tn C t; !// is independent of t for any t D .t1 ; t2 ;    ; tn / 2 T . One of the most important concepts of this paper is that of weakly stationary processes. X.t; !/ is said to be weakly stationary if the following conditions are satisfied. (i) The absolute moment of the second order is finite: EjX.t; !/j2 < 1 for each t 2 T: (ii) The expectation is constant throughout time: EX.t; !/ D m.t/ D m

constant for all t 2 T:

(iii) The covariance depends only upon the difference u D s  t of times : .s; t/ D .s C h; t C h/ for any s; t; h 2 T: The condition (iii) permits us to denote the covariance by .u/ instead of .s; t/ for the sake of simplicity. It is well-known that the strong stationarity implies the weak stationarity provided that EjX.t0 ; !/j2 < 1 for some t0 2 T (cf. Itô [15] pp. 236–237). We add two more concepts, the strong continuity and the measurability, in the case of T D R. Propositions 1 and 2 clarify the relation between these concepts. (See Kawata [19] pp. 53–55 and Crum [8] for details.) If a function A W R ! L2 .˝; C/ defined by A W t 7! X.t; !/ is continuous at a point t0 2 RI i:e: EjX.t; !/  X.t0 ; !/j2 ! 0 as

t ! t0 ;

we say that X.t; !/ is strongly continuous at t0 . If X.t; !/ is strongly continuous at every t 2 R; X.t; !/ is said to be strongly continuous on R. If X.t; !/ is .L ˝ E ; B.C//-measurable, X.t; !/ is called a measurable stochastic process, where L is the Lebesgue -field on R. Proposition 1. Let X W R  ˝ ! C be a weakly stationary process with EX.t; !/ D 0 for all t 2 R.

Periodic Weakly Stationary Processes

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(i) X.t; !/ is strongly continuous at t D 0 if and only if X.t; !/ is uniformly strongly continuous on R. (ii) .u/ is continuous at u D 0 if and only if .u/ is uniformly continuous on R. (iii) X.t; !/ is strongly continuous if and only if .u/ is uniformly continuous on R. Proposition 2. The covariance function of any measurable and weakly stationary process X.t; !/ is continuous. So X.t; !/ is strongly continuous on R.

2 Linear Stochastic Processes One of the most typical examples of weakly stationary processes is so called the linear stochastic process (the Slutsky process, informally). We assume, for a while, the case of discrete time; i.e. T D Z. And, according to the ordinary custom, we write Xn .!/ rather than X.n; !/. Let Xn W ˝ ! C.n 2 Z/ be a stochastic process such that (i) EXn .!/ D 0 for all n 2 Z, (ii) EjXn .!/j2 D  2 ( 2 is a constant, called the variance, which is independent of n), and (iii) EXn .!/Xm .!/ D 0 for all m ¤ n. A stochastic process which satisfies (i), (ii) and (iii) above is called a white noise. The condition (iii) is called the serially uncorrelatedness. The next proposition confirms that some average of a white noise is weakly stationary.3 Proposition 3. Assume that a stochastic process Xn W ˝ ! C.n 2 Z/ satisfies (i), 1 X jc j2 < 1 (i.e. fc g 2 `2 .C/). (ii) and (iii), and fc g2Z is a sequence with Define another stochastic process Yn .!/ by Yn .!/ D

1 X

nD1

ckn Xk .!/;

n 2 Z:

(1)

kD1

Then the right-hand side of (1) is L2 -convergent and Yn .!/.n 2 Z/ is a weakly stationary process with EYn .!/ D 0;

3

cf. Kawata [19] p. 46.

n 2 Z;

(2)

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T. Maruyama

and .u/ D  2

1 X

cuC cN  :

(3)

D1

The weakly stationary process defined by (1) is called the moving average process or the linear stochastic process induced by Xn .!/ .n 2 Z/. Proposition 1 says that a moving average process induced by a white noise is weakly stationary. The corresponding concept of linear stochastic processes in continuous-time case (T D R) will be defined later (cf. Theorem 2 in Sect. 7) because some additional prerequisite is required.

3 Spectral Measures Let T be equal to Z or R. A function f W T ! C is said to be positive semi-definite if n X

f .ti  tj /i Nj = 0

i;jD1

for any finite tuple t D .t1 ; t2 ;    ; tn / 2 T of elements of T and 1 ; 2 ;    ; n 2 C. The celebrated theorems due to Herglotz [14] and Bochner [2, 3] tell us that any positive semi-definite function on T can be expressed as the Fourier transform of certain positive Radon measure.4 Herglotz Theorem. The following two statements are equivalent for a function f W Z ! C.

4 Schwartz [27] Chap. VII, §9, and Lax [20] Chap. 14 discuss the Bochner-Herglotz theorem in the spirit of the theory of distributions. Rudin [25] Chap. 1 provides a proof based upon the theory of Banach algebras. Naimark [24] Chap. 6 gives an abstract version, following D.A. Raikov.

“Let G be a locally compact commutative topological group with unit e. Then any continuous positive semi-definite function ' W G ! C is uniquely representable in the form Z .g/d./; '.g/ D

b G

where  is a measure on the character group b G of G, which satisfies .b G/ D '.e/:” The converse also holds true.

Periodic Weakly Stationary Processes

157

(i) f is positive semi-definite. (ii) There exists a positive Radon measure  on T D Π;  which satisfies 1 f .n/ D p 2

Z

einx d.x/: T

Bochner Theorem. The following two statements are equivalent for a bounded continuous function f W R ! C. (i) f is positive semi-definite. (ii) There exists a positive Radon measure  on R which satisfies 1 f .t/ D p 2

Z

eitx d.x/:

R

Let X W T  ˝ ! C be a weakly stationary stochastic process. Then it is easy to see that its covariance function .u/ is positive semi-definite. In fact, n X i;jD1

.ti  tj /i j D

n X

EX.ti ; !/X.tj ; !/i j

i;jD1 n ˇ2 ˇX ˇ ˇ D Eˇ X.tj ; !/j ˇ = 0 i;jD1

for any tj 2 T and j 2 C . j D 1; 2;    ; n/. Proposition 4. If X W Z  ˝ ! C is a weakly stationary process, its covariance function .u/ can be expressed as the Fourier transform of certain positive Radon measure  on T W Z 1 .n/ D p ein d./: 2 T This is an immediate consequence of the Herglotz Theorem and the positive semi-definiteness of . There remains nothing to be explained more. However we have to check the continuity of  in the case T D R.

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T. Maruyama

Proposition 4’. If X W R  ˝ ! C is a measurable and weakly stationary process, its covariance function .u/ can be expressed as the Fourier transform of certain positive Radon measure  on R W 1 .u/ D p 2

Z

eiut d.t/: R

Since X.t; !/ is measurable, .u/ is continuous by Proposition 2. We already confirmed the positive semi-definiteness of .u/. Hence Proposition 4’ follows from the Bochner Theorem. The Radon measure  appearing Proposition 4 or Proposition 4’ is called the spectral measure of X.t; !/. If  is absolutely continuous with respect to the Lebesgue measure, the Radon-Nikodým derivative is called the spectral density function of X.t; !/. (See Sect. 7.) It is well-known in Fourier analysis that if the Fourier transform O of a Radon measure  on R or T is zero, then  D 0 (cf. Katznelson [16] p. 164). Hence the mapping  7! O is injective. We next consider the converse problem. When a positive Radon measure  on T is given, does there exist a weakly stationary stochastic process the spectral measure of which is exactly ? The following proposition answers this question positively. Proposition 5. A positive Radon measure  on T is assumed to be given. Then there exist a probability space .˝; E ; P/ and a stochastic process Xn W ˝ ! C.n 2 Z/ which satisfies the following conditions. (i) Xn .!/ is a weakly stationary process. (ii)  is the spectral measure of Xn .!/. The next proposition is a continuous-time (T D R) version of the above. Proposition 5’. A positive Radon measure  on R is assumed to be given. Then there exist a probability space .˝; E ; P/ and a stochastic process X W R  ˝ ! C which satisfies the following conditions. (i) X.t; !/ is a measurable and weakly stationary process. (ii)  is the spectral measure of X.t; !/.

4 Spectral Representation of Weakly Stationary Processes We define the subclass B  .R/ of the Borel -field on R by B  .R/ D fS 2 B.R/jm.S/ < 1g. m./ is the Lebesgue measure. .˝; E ; P/ is a probability space.

Periodic Weakly Stationary Processes

159

A function W B.R/  ˝ ! C is called an orthogonal measure (or L2 orthogonal measure) if the following three conditions are satisfied. (i) The function ! 7! .S; !/ is in L2 .˝; C/ for every S 2 B  .R/. 1 [ Sn 2 B  .R/, then (ii) If S1 ; S2 ;    2 B  .R/ are mutually disjoint and nD1

.

1 [

Sn ; !/ D

nD1

1 X

.Sn ; !/ in L2 .˝; C/:

nD1

0

0

(iii) If S; S 2 B  .R/ and S \ S D ;, then E .S; !/ .S 0 ; !/ D 0: If (i), (ii) and (iii) hold true on B.R/ instead of B  .R/, .S; !/ is called a finite orthogonal measure. In this case, the set function  W B.R/ ! R defined by  .S/ D k .S; !/k22 ;

S 2 B.R/

(1)

is a finite measure on (R; B.R/). In fact, it is obvious that  .S/ = 0 for all S 2 B.R/. Furthermore, if Sn 2 B.R/ .n D 1; 2;    / are mutually disjoint, then  .

1 [ nD1

1 n X  [ 2 2     Sn / D  . Sn ; !/ D lim  .Sj ; !/ nD1

D lim

.iii/ n!1

n X jD1

2 .ii/ n!1

k .Sj ; !/k22 D

jD1 1 X

2

 .Sn /:

nD1

This prove the -additivity of  ./. We define the concept of integration with respect to the finite orthogonal measure .S; !/ for functions f 2 L2 .R; C/ (L2 -space w.r.t.  ). We start with defining the integration for simple functions, and then proceed to general L2 -functions based upon extention by continuity. See Brémaud [5] pp. 137–141, Doob [9] Chap. IX and Kawata [18] pp. 28–35 for details.

160

T. Maruyama

It can easily be proved that Z

Z E

R

f .x/ .dx; !/

Z R

g.x/ .dx; !/ D

R

f .x/g.x/d .x/

(2)

for any f ; g 2 L2 .R; C/:

In particular, Z 2 Z   j f .x/j2 d .x/  f .x/ .dx; !/ D R

2

R

for f 2

(3)

L2 .R; C/:

For the sake of later reference, we call (2) and (3) by the name “the Doob-Kawata formulas” (D-K formulas). The next theorem due to Cramér [7] and Kolmogorov5 gives a spectral representation of weakly stationary processes.6 Cramér-Kolmogorov Theorem. Let X W R  ˝ ! C be a measurable and weakly stationary process with EX.t; !/ D a .for all t 2 R). Then there exists an orthogonal measure W B.R/  ˝ ! C which satisfies 1 X.t; !/ D a C p 2

Z R

eit .d; !/

(4)

and  D  . Conversely, the stochastic process X.t; !/ represented by the above formula (4) in terms of an orthogonal measure .S; !/ is weakly stationary. is uniquely determined corresponding to X. The same result holds true for the case T D Z. In this case, B.R/ should be replaced by B.T/ and the scope of integration in (4) should be T instead of R.

5 Periodicity of Weakly Stationary Processes Let X W T  ˝ ! C be a weakly stationary process with the covariance function .u/. X.t; !/ is called a periodic weakly stationary process with period  or, in short, -periodic if the .u/ is periodic with period ; i.e. .u C / D .u/.

5

According to Itô [15] p. 255, Kolmogorov’s important article was published in C.R. Acad. Sci. URSS, 26 (1940), 115–118. However I have never read it yet, very regrettably. That is why I dropped it from the reference list.

In case X.t; !/ is real-valued, is it possible to give an expression of it in terms of an orthogonal measure without complex function like eit ? This probelm was advocated by Slutsky [28, 29] and completed by Doob [10] and Maruyama [23]. See also Itô [15] pp. 263–266.

6

Periodic Weakly Stationary Processes

161 F(α )

Fig. 1 Spectral density function of a periodic process

−2π /τ

0

2π /τ 4π /τ

2kπ /τ

Proposition 6 is a characterization of periodic processes in discrete-time case. Proposition 6’ is its continuous-time version. Proposition 6. Let X W Z˝ ! C be a weakly stationary process with the spectral measure . Then the following three statements are equivalent. (i) Xn .!/ is -periodic. (ii) XnC .!/  Xn .!/ D 0 a:e:.!/ for all n 2 Z. (iii) If E 2 B.T/ and E \ f2k =jk 2 Zg D ;, then .E/ D 0. Proposition 6’. Let X W R˝ ! C be a measurable and weakly stationary process with the spectral measure . Then the following three statements are equivalent. (i) X.t; !/ is -periodic. (ii) X.t C ; !/  X.t; !/ D 0: a:e:.!/ for all t 2 R. (iii) If E 2 B.R/ and E \ f2k =jk 2 Zg D ;; then .E/ D 0. Remark. The spectral distribution function of X.t; !/ is defined by F.˛/ D ..1; ˛/;

˛ 2 R:

Then (iii) means that F.˛/ is a step function with possible discontinuities at f2k =jk 2 Zg. (See Fig. 1) If X.t; !/ is a -periodic weakly stationary process, the spectral measure concentrates on a countable set in T or R, informally called the energy set of X.t; !/, such that the distance of any adjacent two points is some multiple of 2 =. We have to keep in mind that the periodic weakly stationary process can not have spectral density function.

6 Almost Periodicity of Weakly Stationary Processes Let f W R ! C be a function and " a positive real number.  2 R is called an "-almost period of f if sup j f .x  /  f .x/j < ": x2R

162

T. Maruyama

f is said to be (uniformly) almost periodic in the sense of Bohr [4] if (i) f is continuous, and (ii) there exists  D ."; f / 2 R for each " > 0 such that any interval, the length of which is ."; f /, contains "-almost period of f . The set AP.R; C/ of all the almost periodic functions of R into C forms a closed subalgebra of L1 .R; C/.7 Let X W T  ˝ ! C be a weakly stationary process with the covariance function .u/. X.t; !/ is called an almost periodic weakly stationary process if .u/ is almost periodic. The following propositions play similar roles with those of Propositions 6 and 6’ in Sect. 5. Proposition 7. Let X W Z˝ ! C be a weakly stationary process with the spectral measure . Then the following three statements are equivalent. (i) Xn .!/ is an almost periodic process. (ii) There exists  D  ."; X/ 2 R for each " > 0 such that any interval, the length of which is  ."; X/, contains some  2 Z which satisfies sup EjXnC .!/  Xn .!/j2 < ": n2Z

(iii)  is discrete. Proposition 7’. Let X W R˝ ! C be a measurable and weakly stationary process with the spectral measure . Then the following three statements are equivalent. (i) X.t; !/ is an almost periodic process. (ii) There exists  D  ."; X/ 2 R for each " > 0 such that any interval, the length of which is  ."; X/, contains some  2 R which satisfies sup EjX.t C ; !/  X.t; !/j2 < ": t2R

(iii)  is discrete.

7 Spectral Density Functions As we saw in previous sections, any periodic or almost periodic weakly stationary process can not have spectral density functions.

7

See Katznelson [16] p. 194. Loomis [21] is also beneficial.

Periodic Weakly Stationary Processes

163

Hence a problem arises: under what conditions does a weakly stationary process have a spectral density function?8 Let W B.T/  ˝ ! C be an orthogonal measure with E .S; !/ D 0 for any S 2 B.T/. There exist a probability space .˝ 0 ; E 0 ; P0 / and a weakly stationary process Yn .! 0 / W ˝ 0 ! C.n 2 Z/, the spectral measure of which is the Lebesgue measure m on T (by Proposition 5). We denote by E! (resp. E! 0 ) the expectation operator on .˝; E ; P/ (resp. .˝ 0 ; E 0 ; P0 /). The Cramér-Kolmogorov Theorem assures the representation 1 Yn .! 0 / D p 2

Z T

ein .d; ! 0 /

for some orthogonal measure W B.T/˝ 0 ! C. (We assume that E! 0 Yn .! 0 / D 0.)

satisfies (a) E! 0 .S; ! 0 / D 0 for any S 2 B.T/, and (b) E! 0 j .S; ! 0 /j2 D m.S/ for any S 2 B.T/. In order to express some relations between and , we need the “adjunction” method9 as follows. E.!;! 0 / denotes the expectation operator on the product probability space .˝  ˝ 0 ; E ˝ E 0 ; P ˝ P0 /. 1.!/ (resp. 1.! 0 /) denotes the constant function which is identically 1 on ˝ (resp. on ˝ 0 ). Then the following conditions hold true. (i) E.!;! 0 / .S; ! 0 /1.!/ D E! 0 .S; ! 0 / D 0 for any S 2 B.T/. (ii) E.!;! 0 / j .S; ! 0 /1.!/j2 D  .S/ D m.S/ for any S 2 B.T/. (iii) E.!;! 0 / .S; !/1.! 0 /  .S0 ; ! 0 /1.!/ D E! .S; !/E! 0 .S0 ; ! 0 / D 0 and S0 2 B.T/.

for any S

Theorem 1. Let Xn .!/ .n 2 Z/ be a weakly stationary process with the spectral measure . Assume also that EXn .!/ D 0. Thus the following two statements are equivalent. (i) Xn .!/ has the spectral density function. (ii) Xn .!/ is a linear stochastic process; that is there exist a sequence fcn g of complex numbers such that 1 X

jcn j2 < 1

nD1

8

This problem was studied by Doob [9] Chap. X, §8, Chap. XI, §8 and Kawata [19] pp. 69–73. I try to clarify the subtle details embedded in their works.

9

See Doob [9] Chap. II, §2.

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T. Maruyama

and a stochastic process Zn .!/.n 2 Z/ with EZn .!/ D 0;

EjZn .!/j2 D 1;

EZn .!/Zm .!/ D 0

(1)

if n ¤ m

which satisfy 1 X

Xn .!/ D

ckn Zk .!/

a:e:

kD1

(The convergence on the right-hand side is in L2 .˝; C/.) Proof. (i))(ii): Let p./ = 0 be the Radon-Nikodým derivative of . We also define p ˛./ D p./: (2) Since ˛./ 2 L2 .T; C/ (actually real-valued), it can be expanded by the Fourier series10 : 1 X 1 ˛./ D p ˛k eik ; 2 kD1 1 X

j˛k j2 < 1;

(3)

(4)

kD1

p where ˛k is the Fourier coefficient corresponding to .1= 2 /eik .k 2 Z/. By the Cramér-Kolmogorov theorem, Xn .!/ is represented as the Fourier transform of some orthogonal measure .S; !/. According to the argument preceding the theorem, there exist a probability space .˝ 0 ; E 0 ; P0 / and an orthogonal measure

.S; ! 0 / W B.T/˝ 0 ! C which satisfies (i), (ii) and (iii) mentioned above (p. 163). Divide T into TC and T0 defined as TC D ft 2 Tj˛.t/ > 0g; T0 D ft 2 Tj˛.t/ D 0g:

The convergence of the series in (3) is in L2 .T; C/. However the series is, actually, convergent a.e. and is equal to ˛./ according to the Carleson theorem. cf. Carleson [6].

10

Periodic Weakly Stationary Processes

165

We define a couple of functions, ˛1 ./ and ˛2 ./, by ( ˛1 ./ D ( ˛2 ./ D

1 ˛./

on TC ;

0

on T0 ;

0

on TC ;

1

on T0 :

(5)

Furthermore define a function  0 W B.T/  ˝  ˝ 0 ! C by 11 0

Z

0

 .S; !; ! / D

Z

0

˛2 ./ .d; ! 0 /1.!/:

˛1 ./ .d; !/1.! / C S

S

Z D

(6)

Z

˛2 ./ .d; ! 0 /:

˛1 ./ .d; !/ C S

S

For the definition (6) to be possible, we have to check that ˛1 ./ 2 L2 and ˛2 ./ 2 L2 where  .S/ D E! j .S; !/j2

and  .S/ D E! 0 j .S; ! 0 /j2 ;

S 2 B.T/;

respectively. But there is no difficulty in checking it as shown in the following: Z

Z

2

j˛1 ./j d D S

S\TC

Z D

1 p./dm./ ˛./2 dm./ D m.S \ TC / < 1;

(7)

S\TC

Z S

j˛2 ./j2 d D

Z dm./ D m.S \ T0 / < 1: S\T0

In case T0 D ; (and so ˛./ never vanishes), the discussion becomes much easier, since it is enough to define .S; !/ simply by Z 1 .S; !/ D .d; !/ S ˛./

11

for any S 2 B .T/. Clearly  .S/ D m.S/.

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T. Maruyama

 0 .S; !; ! 0 / is an orthogonal measure. For instance, the orthogonality is proved as follows. Let S and S0 2 B.T/ be disjoint. Then we obtain E.!;! 0 /  0 .S; !; ! 0 / 0 .S0 ; !; ! 0 / Z Z D E! ˛1 ./ .d; !/ ˛1 ./ .d; !/ S

S0

Z Z

˛2 ./ .d; ! / ˛2 ./ .d; ! 0 /

S0

Z S0

S

˛2 ./ .d; ! 0 /

(8)

Z

0

S

Z

S0

S

C E.!;! 0 / C E! 0

Z

˛1 ./ .d; !/

C E.!;! 0 /

˛1 ./ .d; !/

˛2 ./ .d; ! 0 /:

The second and third terms of (8) are zero because of the orthogonality of and

in the sense of (iii) (p. 163). The first and the forth terms are also zero because S \ S0 D ;. By a similar computation as in (7) and (8), we have 12 E.!;! 0 / j 0 .S; !; ! 0 /j2 D m.S/;

S 2 B.T/:

(9)

Finally a function  W B.T/  ˝ ! C is defined by .S; !/ D E! 0  0 .S; !; ! 0 /:

(10)

Taking account of the properties of  0 ,  is shown to be an orthogonal measure with  .S/ D m.S \ TC /.

12

E.!;! 0 / j 0 .S; !; ! 0 /j2 ˇZ ˇ2 ˇ ˇ DE! ˇ ˛1 ./ .d; !/ˇ S

ˇZ ˇ2 ˇ ˇ C E! 0 ˇ ˛2 ./ .d; ! 0 /ˇ S

C 2R eE.!;! 0 / Z D S

Z S

˛1 ./ .d; !/1.! 0 /

j˛1 ./j2  .d/ C

C 2R eE.!;! 0 /

Z S\TC

Z S

Z S

˛2 ./ .d; ! 0 /1.!/

j˛2 ./j2  .d/

˛1 ./ .d; !/

Dm.S \ TC / C m.S \ T0 / C 0:

Z S\T0

˛2 ./ .d; ! 0 /

Periodic Weakly Stationary Processes

167

The Cramér-Kolmogorov representation theorem gives 13 1 Xn .!/ D p 2

1 D p 2

Z T

Z T

ein .d; !/ (11) ein ˛./.d; !/:

Since the Fourier series (3) converges to ˛./ in L2 , it follows that Xn .!/ D

1 Z 1 X ˛k ei.nk/ .d; !/ a:e: 2 kD1 T

(12)

In fact, (12) is verified by the computation: p Z ˇ ˇ2 1 X ˇ ˇ E! ˇXn .!/  ˛k ei.kn/ .d; !/ˇ 2 kDp T

Z X p ˇ ˇ2 1 ˇ ˇ D E! ˇXn .!/  ˛k ei.kn/ .d; !/ˇ 2 T kDp p ˇ2 ˇ 1 Z h i 1 X ˇ ˇ D E! ˇ p ein ˛./  p ˛k eik .d; !/ˇ 2 T 2 kDp

1 D 2

13

Z T

Z ˇ p ˇ2 1 X ˇ ˇ ˛k eik ˇ d ./ ˇ˛./  p 2 kDp T

.by (11)/

.by D-K formula/

ein ˛./.d; !/ Z

Z 1 ein ˛./  ˛2 ./ .d; ! 0 / .d; !/ C E! 0 ˛./ TC T0 Z Z D ein .d; !/ D ein .d; !/:

D

ein ˛./ 

TC

T

The final equality is justified by Z ˇZ ˇ2 ˇ ˇ E! ˇ ein .d; !/ˇ D T0

T0

Z

D T0

p./dm D 0

 .d/

(by D-K formula)

. p./ D 0 on T0 /:

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T. Maruyama

D

1 2

5

1 2

Z

p ˇ ˇ2 1 X ˇ ˇ ˛k eik ˇ dm./ ˇ˛./  p 2

TC kDp

Z ˇ p ˇ2 1 X ˇ ˇ ˛k eik ˇ dm./ ˇ˛./  p 2 kDp T

!0

as p ! 1

.by (3)/:

If we define 1 Zj .!/ D p 2

Z T

eij .d; !/;

(13)

p and ck D .1= 2 /˛k , then Xn .!/ D

1 X

1 X

ck Zkn .!/ D

kD1

cnCj Zj .!/

.in L2 /:

(14)

jD1

It is easy to check that Zn .!/ satisfies the condition (1). EZj .!/ D 0 is clear from the definition (10) of  .E D E! /. The varianceD 1 and the orthogonality come from 2 EZn .!/Zm .!/ Z Z in DE e .d; !/ eim .d; !/ T

Z D Z

T

D

Z

ei.nm/ d D

T

ei.nm/ dm./ TC

Z

ei.nm/ dm./  T

T0

Z ei.nm/ d D

ei.nm/ dm./ T

Dın;m  2 : The second equality is justified by D-K formula. (ii))(i): Conversely, assume that Xn .!/ is a moving average of a stochastic process Zn .!/ which satisfies (1). Since Zn .!/ is weakly stationary, there exists an orthogonal measure .S; !/ on B.T/  ˝ such that 1 Zn .!/ D p 2

Z T

ein .d; !/

(15)

Periodic Weakly Stationary Processes

169

by the Cramér-Kolmogorov theorem. If we define  .S/ D k .S; !/k22 as usual, p  is the spectral measure of Zn .!/, which has the spectral density function 1= 2

(constant function). 14 Consequently, we obtain, for some fcn g 2 `2 .C/, that Z N 1 X Xn .!/ D l:i:m: p ckn eik .d; !/ N!1 2

T kDN Z X N 1 D l:i:m: p ckn eik .d; !/: N!1 2 T kDN

(16)

Since fcn g 2 `2 , cn is the Fourier coefficient of some C./ 2 L2 .T; C/, and15 q  1 X    ck eik  C./ ! 0 p 2 2 kDp

as p; q ! 1:

(17)

Hence (writing k  n D j), N X

ckn eik D ein

kDN

tends to

Nn X jDNn

p 2 ein C./ in L2 as N ! 1 for fixed n; i.e. N  X  p   ckn eik  2 ein C./ ! 0  2

kDN

14

15

cj eij

as N ! 1:

(18)

Since Zn .!/ is a white noise, the covariance is given by ( 1 if u D 0; EZnCu .!/Zn .!/ D 0 if u ¤ 0: q X p The convergence of .1= 2 / ck eik to C./ also holds true “almost everywhere” kDp

thanks p to the Carleson theorem. Hence ck is the Fourier coefficient of C./ corresponding to .1= 2 /eik .

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T. Maruyama

p Taking account of the fact that the density function of  is 1= 2 as remarked above, we have Z ˇ ˇ2 ˇ ˇ EˇXn .!/  C./ein .d; !/ˇ T

Z Z X N ˇ ˇ2 1 ˇ ˇ DEˇl:i:m: p ckn eik .d; !/  C./ein .d; !/ˇ N!1 2 T T kDN

.by .16//

Z Nn ˇ2 i p 1 ˇˇ h X ˇ Eˇ D lim cj eij  2 C./ ein .d; !/ˇ N!1 2

T jDNn 1 N!1 2

D lim

Z ˇ Nn ˇ2 p ˇ X ˇ cj eij  2 C./ˇ d ˇ T

Z ˇ ˇ 1 ˇp N!1 T 2

D lim D0

.by D-K formula/

jDNn

ˇ2 1 ˇ cj eij  C./ˇ p dm./ 2

jDNn Nn X

.by .18//:

Hence we obtain 1 Xn .!/ D p 2

Z p 2 C./ein .d; !/:

a:e:

T

If we define .S; !/ by .S; !/ D

Z p

2 C./ .d; !/;

S 2 B.T/;

S

then .S; !/ is an orthogonal measure16 and 1 Xn .!/ D p 2

16

Z T

ein .d; !/:

The orthogonality, for instance, can be verified as follows. If S and S0 2 B .T/ are disjoint, Z Z E.S; !/.S0 ; !/ D E C./S ./ .d; !/ C./S0 ./ .d; !/ Z D T

T

T

jC./j2 S ./S0 ./d D 0

.by D-K formula/:

Periodic Weakly Stationary Processes

171

This is the spectral representation of Xn .!/ in terms of .S; !/. Consequently the spectral measure  of Xn .!/ is given by .S/ D Ej.S; !/j2 D 2

Z

jC./j2 d S

D

p Z 2 jC./j2 dm./; S

which is, of course, absolutely continuous with respect to m.

Q.E.D.

We prepare a lemma to be used in the proof of Theorem 2. We denote by F2 (resp. F1 2 ) the Fourier transform (resp. inverse Fourier transform) in the sense of Plancherel.17 Lemma 1. 1 p 2

Z

1 1 f ./g./eiu dm./ D p hF1 N ./i 2 f .  u/; F2 g 2

R Z 1 Dp f .  u/g./dm./ 2 R for any f and g 2 L2 .R; C/:

(We have to note f  g 2 L1 .R; C/. h; i denotes the inner product in L2 .R; C/.) Proof. By definition of the inverse Fourier transform in the sense of Plancherel, we have izu F1 /./ 2 . f .z/e

1 D l:i:m: p A!1 2

1 D l:i:m: p A!1 2

Z Z

A A

f .z/eizu eiz dm.z/

A A

f .z/ei.u/z dm.z/

D F1 2 f .  u/:

17

We can also establish 1 p 2

Z R

1 f ./g./eiz dm./ D p .F2 f  F2 g/.z/: 2

(19)

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T. Maruyama

Taking account of f  g 2 L1 , we obtain 1 p 2

Z R

f .z/  g.z/eizu dm.z/ 1 D p h f .z/eizu ; gN .z/i 2

1 izu D p hF1 /./; F1 N ./i 2 . f .z/e 2 g 2

(by Parseval equality) 1 1 D p hF1 N ./i (by (19) 2 f .  u/; F2 g 2

1 D p h f .  u/; gN ./i (by Parseval) 2

Z 1 f .  u/g./dm./: D p 2 R Q.E.D.

Theorem 2. Let X.t; !/.t 2 R/ be a measurable and weakly stationary process with the spectral measure . Assume also that EX.t; !/ D 0. Then the following two statements are equivalent. (i) X.t; !/ has the spectral density function. (ii) There exists a measurable and weakly stationary process X 0 W R  ˝ ! C of the form X 0 .t; !/ D

Z R

w.  t/.d; !/ a:e: .!/;

the spectral measure of which is , where w./ 2 L2 .R; C/ and the orthogonal measure .S; !/ W B.R/  ˝ ! C satisfies  .S/ D Ej.S; !/j2 D m.S \ R0 /; S 2 B  .R/: (The definition of R0 is given below.)

Periodic Weakly Stationary Processes

173

Proof. By the Cramér-Kolmogorov theorem, there exists an orthogonal measure .S; !/ on B.R/  ˝ such that Z

1 X.t; !/ D p 2

R

eit .d; !/:

(i))(ii): Let p./ = 0 be the Radon-Nikodým derivative of . If we define ˛./ D

p p./;

then ˛ 2 L2 .R; R/. The covariance .u/ of X.t; !/ can be written as Z Z 1 1 iu .u/ D p e d./ D p eiu ˛./  ˛./dm./ (20) 2 ZR 2 R Z 1 1 ˛.  u/˛./dm./ D p ˛.  u/˛./dm./: D p 2 R 2 R The third equality is assured by Lemma 1. Divide R into RC and R0 defined as RC D ft 2 Rj˛.t/ > 0g; R0 D ft 2 Rj˛.t/ D 0g. As in Theorem 1, we introduce the orthogonal measure .S; !/ W B.R/  ˝ ! C, two functions ˛1 ./ and ˛2 ./, and another orthogonal measure .S; !/. We can construct and  so as to be  .S/ D m.S/;

for S 2 B  .R/

 D m.S \ RC /

.see section 4/:

If we define a weakly stationary process X 0 .t; !/ by 1 X 0 .t; !/ D p 4 2

Z R

˛.  t/.d; !/;

then the covariance 0 .u/ of X 0 .t; !/ is18 0 .u/ D EX 0 .t C u; !/X 0 .t; !/ Z Z 1 D p E ˛.  .t C u//.d; !/ ˛.  t/.d; !/ 2

R R 18

The third line of (21) Z Z o 1 n D p ˛.  .t C u//˛.  t/dm./ C ˛.  .t C u//˛.  t/ d ƒ‚ … 2 RC R0 „ Z Z Z o 1 n D p ./dm./  ./dm./ C ./d 2 R R0 R0

./

(21)

174

T. Maruyama

Z 1 D p ˛.  .t C u//˛.  t/d ./ 2 R Z 1 D p ˛.  u/˛./dm./: 2 R

.by D-K formula/

By (20) and (21), we get  D 0 . That is, the spectral density function of X 0 .t; !/ is p./. (ii))(i): Conversely, assume (ii). The covariance .u/ of Z X.t; !/ D

R

w.  t/.d; !/

is given by Z .u/ D

1

w.  u/w./dm./ 1

as in the calculation of 0 ./ above. Then we get, again by Lemma 1, Z w.  u/w./dm./

.u/ D Z

R

jw./j2 eiu dm./

D R

1 Dp 2

Z p 2 jw./j2 eiu dm./: R

p Finally, if we define p./ D 2 jw./j2 , p./ is the spectral density function of X.t; !/ . Q.E.D.

1 n D p 2

 1 D p 2

Z R

./dm./

Z

Z

2R0 t2RC

nZ

R

./dm./ 

2R0 t2R0

Z ./dm./ C

Z 2R0 t2RC

./d ./ C

2R0 t2R0

o ./d ./

./dm./

Z

 .R0 t/\RC

˛.0  u/˛.0 /dm.0 / C

The last two terms cancel out. So we obtain (21).

Z .R0 t/\RC

o ˛.0  u/˛.0 /d .0 / :

Periodic Weakly Stationary Processes

175

8 Conclusion A (measurable) weakly stationary process X.t; !/ is -periodic if and only if its spectral measure  concentrates on a countable set in T or R such that the distance of any adjacent two points is some multiple of 2 = (cf. Proposition 6, 6’). X.t; !/ is almost periodic if and only if  is discrete (cf. Proposition 7, 7’). Therefore if X.t; !/ is periodic or almost periodic, the spectral measure  can never be absolutely continuous with respect to the Lebesgue measure. Now under what conditions does the spectral measure have the density function? The answers were given in Theorems 1 and 2.  is absolutely continuous with respect to the Lebesgue measure if and only if X.t; !/ is a moving average process of white noise. So we can conclude that any moving average process can neither be periodic nor almost periodic. n ˇ nX o ˇ However we know that the set MdC .R/ D ˛j ıxj ˇ˛j 2 R; xj 2 R; n 2 N of jD1

all the discrete positive Radon measures is w -dense in MC .R/, the set of all the positive Radon measures (cf. Billingsley [1], Malliavin [22] Chap. II, §6). The following Theorem immediately follows from this observation. Theorem 3. Let X.t; !/ be a measurable and weakly stationary process on R  ˝ with the spectral measure . Then there exists a sequence of almost periodic weakly stationary process X k .t; !/ with the spectral measure  k such that  k converges to  in the w -topology. Proof. Since MdC .R/ is w -dense in MC .R/ (metrizable), there exists a sequence  k 2 MdC .R/ which w -converges to . By Proposition 5, there is a measurable and weakly stationary process X k .t; !/, the spectral measure of which is exactly equal to  k . X k .t; !/ is almost periodic because  k 2 MdC .R/. Q.E.D. A similar result also holds true for discrete time case. (In this case R should be replaced by T as usual.) Theorem 3 tells us that any weakly stationary process X.t; !/ can be approximated by a sequence fX k .t; !/g of almost periodic weakly stationary processes in the sense that the sequence of the spectral measures of X k .t; !/ converges to the spectral measure of X.t; !/ in the w -topology.

Appendix We reviewed, in the first few sections, some basic concepts and results to be required in our main concerns (Theorems 1, 2, and 3). I expect this exposition to be a small help for the readers who are not familiar with them. There can be no communications without common language. Of course, these materials are, more or

176

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less, known to every mathematician working in this discipline. That is why we just stated necessary facts and dropped proofs in the text. However it does not seem to be a waste of time to give here complete proofs of selected three propositions (Propositions 5’, 6’ and 7’) as exceptions, taking account of their indispensable roles in our problem. Proof of Proposition 5’. The proof is basically due to Kawata [19] pp. 56–57.19 If we define  W B.R/ ! R by .E/ D

.E/ ; .R/

E 2 B.R/;

then  is a Radon probability measure. There exist independent real random variables Z.!/ and Y.!/ defined on some probability space .˝; E ; P/ such that the p distribution of Z.!/ is equal to , EY.!/ D 0, and finally EY.!/2 D .1= 2 /.R/. 20 Define a stochastic process X.t; !/ by X.t; !/ D Y.!/eiZ.!/t : Then EX.t; !/ and .t C u; t/ are calculated as follows.21 EX.t; !/ D EY.!/EeiZ.!/t D 0 (by independence); .t C u; t/ D EX.t C u; !/X.t; !/ D EŒ y.!/2 eiZ.!/u  Z Z 1 1 D p .R/ eiu d./ D p eiu d./: 2

2 R R Thus the covariance .t Cu; t/ of X.t; !/ depends only upon u, and it is expressed as the Fourier transform of . Since the Fourier transform of  is uniformly continuous, X.t; !/ is strongly continuous on R by Proposition 1. The measurability of X.t; !/ is obvious. Q.E.D.

However we have to be more careful about a couple of subtle reasonings. (a) .˝; E ; P/ can not be fixed a priori. It must be chosen suitably. (b) A justification must be given for the construction of independent random variables.

19

20 Let ˚1 ; ˚2 ;    be a sequence of Borel probability measures on R. Then there exists a sequence of independent random variables defined on some probability space .˝; E ; P/, the distributions of which are ˚1 ; ˚2 ;    (cf. Itô [15] p. 68).

Let X1 ; X2 ;    be a sequence of independent real-valued random variables, and g1 ; g2 ;    W R ! C Borel measurable functions. Then Y1 D g1 .X1 /; Y2 D g2 .X2 /;    are also independent random variables (cf. Itô [15] p. 66). So Y.!/ and eiZ.!/t in the text are independent.

21

Periodic Weakly Stationary Processes

177

Proof of Proposition 6’22 . (i)) (ii): Assume that .u/ is -periodic. We then prove that EjX.t C ; !/  X.t; !/j2 D 0 which is equivalent to (ii). By direct computation, we have EjX.t C ; !/  X.t; !/j2 D EjX.t C ; !/j2 C EjX.t; !/j  2Re EX.t C ; !/X.t; !/ D 2.0/  2Re./ D 2..0/  Re.// D0

.by (i)/:

(ii)) (i): Assume (ii). then we have j.u C /  .u/j2 D jEŒX.u C ; !/X.0; !/  X.u; !/X.0; !/j2 D 0:

.by (ii)/

(i)) (iii): If .u/ is -periodic, then we have 2 0 D 2.0/  ./  ./ D p 2

Z R

.1  cos t/d.t/:

Taking account of 1  cos t = 0, we must have .E/ D 0 for any E 2 B.R/ such that E \ ft 2 Rj1  cos t D 0g D ;; which is equivalent to E \ f2k =jk 2 Zg D ;: (iii)) (i): By definition of the spectral measure, 1 .u/ D p 2

22

We follow Kawata [19] pp. 75–76.

Z

eiut d.t/: R

178

T. Maruyama

So putting ak D .f2 k=g/ .k 2 Z/, we obtain, by (iii), that 1 X iu2 k= .u/ D p : ak e 2

This is clearly -periodic.

Q.E.D.

The following well-known lemma23 will be used in the course of the proof of Proposition 7’. Lemma. Let  be a Radon measure on R. If the Fourier transform O is almost periodic, then  is discrete. Proof of Proposition 7’24 . (i))(ii): Assume that ./ is almost periodic. Then we have sup EjX.t C ; !/  X.t; !/j2 D 2Œ.0/  Re./ t2R

D 2ReŒ.0/  ./ 5 2j.0/  ./j

(1)

5 2 sup j.u C /  .u/j: u2R

Since ./ is almost periodic, there exists a number ."=2; / > 0, for each " > 0, such that any interval the length of which is ."=2; / contains an "=2- almost period, . Since f 2 Rj sup j.u C /  .u/j < u2R

" g 2

f 2 Rj sup EjX.t C ; !/  X.t; !/j2 < "g t2R

by (1), we see that (ii) is satisfied by setting  ."; X/ D ."=2; /. (ii))(i): Assume (ii). By a simple calculation, we obtain the inequality j.u C /  .u/j2 D jEŒX.u C ; !/X.0; !/  X.u; !/X.0; !/j2 5 EjX.u C ; !/  X.u; !/j2 EjX.0; !/j2 (by Schwarz inequality): It follows that j.u C /  .u/j 5 ŒEjX.t C ; !/  X.u; !/j2 1=2 .0/1=2 :

23

Katznelson [16] p. 197.

24

The proof of (i),(ii) is due to Kawata [19] pp. 80–82.

Periodic Weakly Stationary Processes

179

Hence we have ˇ n "2 o ˇ  2 Rˇ sup EjX.u C ; !/X.u; !/j2 < .0/ u2R f 2 Rj sup j.u C /  .u/j < "g: u2R

Thus (i) holds true by setting ."; / D  ."2 =.0/; X/. (i)) (iii): This is a direct consequence of the lemma. The covariance function  is represented by the Fourier transform of some positive Radon measure  on R (Proposition 4’). Since .u/ D .u/ O is almost periodic,  must be discrete by the lemma. (iii)) (i): Assume that the spectral measure  of X is discrete, say D

1 X

a n ı n

.ı n W Dirac measure/:

nD1

Then the covariance can be expressed as 1 .u/ D p 2

Z

1 1 X eiu d./ D p an ei n u : 2

R nD1

(2)

Since an = 0, the series (2) is absolutely and so uniformly convergent. Thus .u/ is the uniform limit of trigonometric polynomials,25 and hence almost periodic. Q.E.D.

References 1. Billingsley P (1968) Convergence of probability measures. Wiley, New York 2. Bochner S (1932) Vorlesungen über Fouriersche Integrale. Akademische Verlagsgesellschaft, Leipzig 3. Bochner S (1933) Monotone Funktionen, Stieltjessche Integrale und harmonische Analyse. Math Ann 108:378–410 4. Bohr H (1932) Fastperiodische Funktionen. Springer, Berlin 5. Brémaud P (2014) Fourier analysis and stochastic processes. Springer, Cham

25

A function of the form f .x/ D

n X

aj ei j u

. j 2 R/

jD1

is called a trigonometric polynomial. Any trigonometric polynomial is almost periodic.

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6. Carleson L (1966) On convergence and growth of partial sums of Fourier series. Acta Math 116:135–157 7. Cramér H (1940) On the theory of stationary random processes. Ann Math 41:215–230 8. Crum MM (1956) On positive-definite functions. Proc Lond Math Soc (3) 6:548–560 9. Doob JL (1953) Stochastic processes. Wiley, New York 10. Doob JL (1949) Time series and harmonic analysis. In: Proceedings of Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley, pp 303– 393 11. Frisch R (1933) Propagation problems and inpulse problems in dynamic economics. In: Economic essays in Honor of Gustav Cassel. Allen and Unwin, London 12. Granger CWJ, Newbold P (1986) Forecasting economic time series, 2nd edn. Academic, Orlando 13. Hamilton JD (1994) Time series analysis. Princeton University Press, Princeton 14. Herglotz G (1911) Über Potenzreihen mit positiven reellen Teil in Einheitskreis. Berichte Verh Säcks. Akad Wiss Leibzig Math Phys Kl 63:501–511 15. Itô K (1953) Probability theory. Iwanami, Tokyo (in Japanese) 16. Katznelson Y (2004) An introduction to harmonic analysis, 3rd edn. Cambridge University Press, Cambridge 17. Kawata T (1966) On the Fourier series of a stationary stochastic process I. Z Wahrsch Verw Gebiete 6:224–245 18. Kawata T (1969) On the Fourier series of a stationary stochastic process II. Z Wahrsch Verw Gebiete 13:25–38 19. Kawata T (1985) Stationary stochastic processes. Kyoritsu, Tokyo (in Japanese) 20. Lax PD (2002) Functional analysis. Wiley, New York 21. Loomis LH (1960) The spectral characterization of a class of almost periodic functions. Ann Math 72:362–368 22. Malliavin P (1995) Integration and probability. Springer, New York 23. Maruyama G (1949) The harmonic analysis of stationary stochastic processes. Mem Fac Sci Kyushu Univ Ser A 4:45–106 24. Naimark MA (1972) Normed algebras. Wolters Noordhoff, Groningen 25. Rudin W (1962) Fourier analysis on groups. Wiley, New York 26. Sargent TJ (1979) Macroeconomic theory. Academic, New York 27. Schwartz L (1966) Théorie des distributions. Hermann, Paris 28. Slutsky E (1937) Alcune proposizioni sulla teoria delle funzioni aleatorie. Giorn Inst Ital degli Attuari 8:193–199 29. Slutsky E (1938) Sur les fonctions aléatoires presque périodiques et sur la décomposition des fonctions aléatoires stationnaires en composantes. Actualités Sci Ind 738:38–55 30. Slutsky E (1937) The summation of random causes as the source of cyclic processes. Econometrica 5:105–146 31. Wold H (1953) A study in the analysis of stationary time series, 2nd edn. Almquist and Wicksell, Uppsala

The 6th Conference on Mathematical Analysis in Economic Theory

Date: January 26(Mon)– 29(Thu), 2015 Venue: Lecture Hall, East Research Building, Keio University 2-15-45 Mita, Minato-ku, Tokyo 108-8345, JAPAN organized by The Japanese Society for Mathematical Economics cosponsored by Keio Economic Society, The Oak Society, Inc.

Programme January 26 (Monday)

*Speaker

Morning

9:00–10:00

10:00–11:00 11:10–12:10

Chair: Takuji Arai (Keio University) Keita Owari (The University of Tokyo) On the Lebesgue property and related regularities of monotone convex functions on Orlicz spaces Shigeo Kusuoka (The University of Tokyo) Expectation of diffusion process with absorbing boundary Robert Anderson* (UC Berkeley), L.R. Goldberg, N. Gunther The cost of financing leveraged US equity through futures

© Springer Science+Business Media Singapore 2016 S. Kusuoka, T. Maruyama (eds.), Advances in Mathematical Economics Volume 20, Advances in Mathematical Economics, DOI 10.1007/978-981-10-0476-6

181

182

The 6th Conference on Mathematical Analysis in Economic Theory

Afternoon

13:30–14:30

14:30–15:30

16:00–17:00

17:00–18:00

Chair: Ryozo Miura (Hitotsubashi University) Takuji Arai (Keio University) Local risk-minimization for Barndorff-Nielsen and Shephard models Katsumasa Nishide (Yokohama National University) Heston-type stochastic volatility with a Markov switching regime Chair: Ken Urai (Osaka University) Hisashi Inaba (The University of Tokyo) Recent developments of the basic reproduction number theory in population dynamics Takashi Suzuki* (Meiji Gakuin University), Nobusumi Sagara Exchange economies with infinitely many commodities and a saturated measure space of consumers

January 27 (Tuesday) Morning

9:00–10:00 10:00–11:00

11:10–12:10

Chair: Takashi Suzuki (Meiji Gakuin University) Yuhki Hosoya (Kanto Gakuin University) The NLL axiom and integrability theory Nobusumi Sagara (Hosei University) An indirect method of nonconvex variational problems in Asplund spaces : the case for saturated measure spaces Ali Khan* (Johns Hopkins University), Yongchao Zhang On pure-strategy equilibria in games with correlated information

Afternoon

13:30–14:30 14:30–15:30

16:00–17:00 17:00–18:00

Chair: Hidetoshi Komiya (Keio University) Hiroyuki Ozaki (Keio University) Upper-convergent dynamic programming Vladimir Tikhomirov (Moscow State University) “Problems of the theory of extremal problems and applications” Chair: Ali Khan (Johns Hopkins University) Alexander Ioffe (Israel Institute of Technology) On curves of descent Arturo Kohatsu-Higa (Ritsumeikan University) The probabilistic parametrix method as a simulation method (tentative)

The 6th Conference on Mathematical Analysis in Economic Theory

183

January 28 (Wednesday) Morning

9:00–10:00

10:00–11:00

11:10–12:10

Chair: Kazuya Kamiya (The University of Tokyo) Yoshiyuki Sekiguchi (Tokyo University of Marine Science and Technology) Real algebraic methods in optimization Ronaldo Carpio (University of International Business & Economics) Fast Bellman iteration: An application of Legendre-Fenchel duality to infinite-horizon dynamic programming in discrete time Takashi Kamihigashi (Kobe University) Extensions of Fatou’s lemma and the dominated convergence theorem

Poster Session 12:10–13:30

Takeshi Ogawa (Hiroshima Shudo University) Intermediate goods with Leontief ’s model and joint production with activity analysis in Ricardian comparative advantage Satoru Kageyama (Osaka University), Ken Urai Fiscal stabilization policy in a Phillips model with unstructured uncertainty Hiromi Murakami (Osaka University), Ken Urai Replica core equivalence theorem: an extension of Debreu-Scarf limit theorem to double infinity monetary economies Ryonfun Im (Kobe University), Takashi Kamihigashi An equilibrium model with two types of asset bubbles

Afternoon

13:30–14:30 14:30–15:30

16:00–17:00 17:00–18:00

18:30–21:00

Chair: Shinichi Suda (Keio University) Makoto Hanazono (Nagoya University) Procurement auctions with general price-quality evaluation Chiaki Hara* (Kyoto University), Kenjiro Hirata Dynamic inconsistency in pension fund management Chair: Shigeo Kusuoka (The University of Tokyo) Chia-Hui Chen (Kyoto University) A tenure-clock problem: evaluation, deadline, and up-or-out Nozomu Muto* (Yokohama National University), Shin Sato Bounded response and Arrow’s impossibility Reception at Tsunamachi Mitsui Club

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The 6th Conference on Mathematical Analysis in Economic Theory

January 29 (Thursday) Satellite Session Morning

9:00–9:30

9:30–10:00 10:00–10:30

11:00–11:30

11:30–12:00

Chair: Hiroyuki Ozaki (Keio University) Masayuki Yao (Keio University) Recursive utility and dynamic programming under the assumption of upper convergence: an order-theoretic approach Chaowen Yu (Keio University) Locally robust mechanism design Hiromi Murakami* (Osaka University), Ken Urai Replica core equivalence theorem: an extension of Debreu-Scarf limit theorem to double infinity monetary economies Kohei Shiozawa* (Osaka University), Ken Urai A generalization of social coalitional equilibrium for multiple coalition structures Takeshi Ogawa (Hiroshima Shudo University) Intermediate goods with Leontief ’s model and joint production with activity analysis in Ricardian comparative advantage

Robert Anderson

Takashi Suzuki

Yuhki Hosoya

Nobusumi Sagara

The 6th Conference on Mathematical Analysis in Economic Theory

Ali Khan

Chiaki Hara

Vladimir Tikhomirov, Toru Maruyama, Alexander Ioffe

185

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The 6th Conference on Mathematical Analysis in Economic Theory

Saying “Good-bye" at the reception (Tsunamachi Mitsui Club)

Index

A Absolute moment of the second order, 154 Adjunction’method, 163 Almost periodic, 162 weakly stationary process, 162 Archimedean, 93 Aumann-Bochner fractional, 57 Aumann integral, 35 Aumann-Pettis fractional, 56 Aumann’s equivalence theorem, 105

B Barndorff-Nielsen and Shephard models, 3–21 Behavioral strategy, 113 Bochner-integrable function, 33 mappings, 25 Bochner theorem, 157–158 Bolza problem, 137 Boundary value problem, 24 ˛;1 Œ0; 1/-solution to the, 43 WB;E Bounded marginal rate of substitution, 106

C Carathéodory integrand, 70 mapping, 52 Cardinality of the set of agents, 111 Carleson theorem, 164 Character group/daul group, 156 Coalitional formulation, 106

Commodity characteristics, 105 Compact metric space, 105 Construction of fields, 145–146 Core, 105 Covariance, 154 Cramér-Kolmogorov theorem, 160

D Demand function, 86 Dierentiated commodities, 105 Differentiability, 132 Dirac measures, 108 Distributionalized formulation, 106 Doob-Kawata formulas, 160

E Eberlein-Smulian theorem, 35, 44 "-almost period, 161 Euler equation, 137 Existence in calculus of variations, 147–148 Expectation, 154 Expenditure function, 86

F Fatou’s lemma, 107 Finite convergence property, 94 Finite orthogonal measure, 159 Föllmer-Schweizer decomposition, 11 Fourier transform (resp.inverse Fourier transform), in the sense of Planchere, 171

© Springer Science+Business Media Singapore 2016 S. Kusuoka, T. Maruyama (eds.), Advances in Mathematical Economics Volume 20, Advances in Mathematical Economics, DOI 10.1007/978-981-10-0476-6

187

188 Fractional bochner integral, 25–28 Fractional differential equation (FDE), 24, 67 solutions set .(SZ /, 68 ˛;1 WB;E .Œ0; 1/ solutions set to the, 33, 39, 68, 71, 72 Fractional differential inclusion (FDI), 24, 33–38, 65 ˛;1 .Œ0; 1/ solutions set to the, 33, 39, 41, WB;E 45, 47 Fractional fuzzy differential inclusion, 61–63 Fractional Pettis integral, 28–33 Fractional w-R.L derivative of order, 24, 25, 30, 38 Free Abelian group, 107 Fundamental bifurcation, 113

G Gamma function, 28 Gamma-OU, 6 Gelfand integration, 107 Global solution, 78

H Herglotz Theorem, 156–157 Household economics, 105

I Ideal, 92 IG-OU, 6 Imperfect competition, 104 Individualistic formulation, 106 Individualized economy, 116 Infinitesimal amounts, 105 Inner product, 91 Integrable, 79 selections, 36 Integrably bounded mapping, 61 multifunction, 43 multimapping, 58

K Kakutani-Ky Fan fixed point theorem, 37, 51, 56, 60

L Lagrange problem of calculus of variations, 142 Large economy, 105

Index Lasserre’s hierarchy, 93 Lebesgue-measurable, 29 Lemma on non triviality of annihilator, 135 Level sets, 61 Leverage effect, 4 Locally compact commutative topological group, 156 Lotteries, 112 Lyapunov’s theorem, 107 M Mackey topology, 30, 37, 46, 51 Malliavin calculus, 13 Measurable selections, 52 Minimal martingale measure (MMM), 5 Moment problem, 93 Monopoly power, 104 Moving average process, 156 Multivalued Aumann-Pettis integral, 57 Multivalued fractional Aumann-Pettis integral, 57 Multivalued fractional Pettis integral, 56–67 inclusion, 52 Multivalued Pettis integral inclusion, 52, 54 N Narrow topology, 67 Nonextendable solution, 79 Nonnegative orthant, 78 Nonstandard analysis, 113 O Optimal control problem, 144–145 Ordinary dierential equation (ODE), 24 Ornstein-Uhlenbeck process, 4 Orthogonal measure, 159 P Partial differential equation, 77–87 Periodic weakly stationary process, 160 Pettis-integrable, 28 functions, 28, 38 mapping, 30 Pettis norm, 28 Polynomial optimization problems, 90 Positive orthant, 78 Positive semi-definite, 156 Q Quadratic module, 92

Index R Randomized choice, 113 Real radical, 92 Registers, 113 Relatively weakly compact, 33 Riemann-Liouville fractional derivative, 25 Right-inverse mapping theorem, 133–134

S Saturated economy, 116 Saturated (super-atomless) measure space, 107 saturated measure spaces (Homogeneous), 115 Scalarly integrable function, 28 Scalarly sequentially upper semicontinuous, 36, 60 Scalarly uniformly integrable, 28–29 Scalarly upper semicontinuous, 58  .L1E ; L1 E /-compact set, 45  .P1E ; L1 ˝ E /-convergence, 40 Second order ordinary dierential equation (SODE), 24 Semialgebraic set, 90 Semidefinite programming problem, 91 Separability, 133 Serially uncorrelatedness, 156 Slutsky process, 152 Smooth-convex Lagrange principle, 140–142 Smooth problems, 142–143 Space of signed measures, 105 Spectral density function, 158 Spectral measure, 158 Stable topology, 67, 72 Stochastic volatility, 4 Strassen theorem, 57 Sum of square polynomials (SOS), 91 relaxation, 93

189 Symmetrization, 114 System of finite dimensional distributions, 154

T  .E ; E/, 37 Topology, 175 of pointwise convergence on L1 ˝ E 39, 40 of uniform convergence, 45 Trigonometric polynomial, 179 Truncated ideal, 92 Truncated quadratic module, 92

U Uniformly good substitutes, 110 Uniformly integrable, 28 Upper hemi-continuous, 116 Utility function, 86

V Vanishing ideal, 92 Variance, 155 Variety, 92 Volatility risk premium, 4

W Walrasian general equilibrium theory, 104 Walras’ law, 84 Weakly compact, 36

Y Young measures, 24, 67, 68, 72 control, 67, 68