Stochastic Calculus in Infinite Dimensions and SPDEs (SpringerBriefs in Mathematics) [2024 ed.] 3031695852, 9783031695858

Table of contents :
Preface
Acknowledgements
Contents
1 Introduction
1.1 Motivation and Description of the Brief
1.2 Notation
2 Stochastic Calculus in Infinite Dimensions
2.1 A Classical Construction for Hilbert Space Valued Processes
2.2 Martingale and Local Martingale Integrators
2.3 Cylindrical Brownian Motion
2.4 Martingale Theory in Hilbert Spaces
2.5 Integration with Respect to Cylindrical Brownian Motion
3 Stochastic Differential Equations in Infinite Dimensions
3.1 The Stratonovich Integral
3.2 Strong Solutions in the Abstract Framework
3.3 Uniqueness and Maximality
3.4 Stratonovich SPDEs in the Abstract Framework
3.5 Weak Solutions in the Abstract Framework
3.6 Time-Dependent Operators
4 A Toolbox for Nonlinear SPDEs
4.1 Existence and Uniqueness in Finite Dimensions
4.2 Tightness Criteria
4.3 Cauchy Criteria
4.4 Enhanced Regularity and an Energy Equality
4.5 SPDEs with Constant Multiplicative Noise
Appendix A
A.1 Classical Results from the Real Valued Theory
A.2 Classical Tightness Criteria
A.3 Stochastic Grönwall Lemma
References
Index

Citation preview

SpringerBriefs in Mathematics Daniel Goodair · Dan Crisan

Stochastic Calculus in Infinite Dimensions and SPDEs

SpringerBriefs in Mathematics Series Editors Nicola Bellomo, Torino, Italy Michele Benzi, Pisa, Italy Palle Jorgensen, Iowa City, USA Roderick Melnik, Waterloo, Canada Otmar Scherzer, Linz, Austria Benjamin Steinberg, New York, NY, USA Lothar Reichel, Kent, USA Yuri Tschinkel, New York, NY, USA George Yin, Detroit, USA Ping Zhang, Kalamazoo, MI, USA

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Daniel Goodair • Dan Crisan

Stochastic Calculus in Infinite Dimensions and SPDEs

Daniel Goodair Imperial College London London, UK

Dan Crisan Department of Mathematics Imperial College London, UK

ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-031-69585-8 ISBN 978-3-031-69586-5 (eBook) https://doi.org/10.1007/978-3-031-69586-5 Mathematics Subject Classification: 60H15, 60H05, 60G44, 35R60, 35R15, 35Q35, 60H30 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland If disposing of this product, please recycle the paper.

Preface

The purpose of this brief is to cover the basics of infinite dimensional stochastic differential equations (defined on Hilbert spaces), in a pedagogical manner, assuming only an elementary understanding of functional analysis and probability theory. We present a robust construction of the stochastic integral in Hilbert Spaces, considering integrals driven at first by real valued martingales and later by Cylindrical Brownian Motion, introducing this concept and expanding into a basic set-up for Stochastic Partial Differential Equations (SPDEs). The framework that we establish facilitates a broad class of SPDEs and noise structures, notably including unbounded noise, in which we build upon standard Stochastic Differential Equation (SDE) theory and rigorously deduce a conversion between their Stratonovich and Itô forms. In the remainder of the brief, we explore more advanced tools to be used in the analysis of these equations. London, UK July 2024

Daniel Goodair Dan Crisan

v

Acknowledgements

Daniel Goodair was supported by the Engineering and Physical Sciences Research Council (EPSCR) Project 2478902. Dan Crisan was partially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (ERC, Grant Agreement No 856408).

vii

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation and Description of the Brief. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4

2

Stochastic Calculus in Infinite Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 A Classical Construction for Hilbert Space Valued Processes . . . . . . . 2.2 Martingale and Local Martingale Integrators . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Cylindrical Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Martingale Theory in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Integration with Respect to Cylindrical Brownian Motion . . . . . . . . . . .

7 7 15 23 27 45

3

Stochastic Differential Equations in Infinite Dimensions. . . . . . . . . . . . . . . . 3.1 The Stratonovich Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Strong Solutions in the Abstract Framework. . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Uniqueness and Maximality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Stratonovich SPDEs in the Abstract Framework . . . . . . . . . . . . . . . . . . . . . 3.5 Weak Solutions in the Abstract Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Time-Dependent Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 59 63 71 77 80

4

A Toolbox for Nonlinear SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1 Existence and Uniqueness in Finite Dimensions. . . . . . . . . . . . . . . . . . . . . . 83 4.2 Tightness Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.3 Cauchy Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.4 Enhanced Regularity and an Energy Equality. . . . . . . . . . . . . . . . . . . . . . . . . 104 4.5 SPDEs with Constant Multiplicative Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

ix

x

Contents

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Classical Results from the Real Valued Theory . . . . . . . . . . . . . . . . . . . . . . . A.2 Classical Tightness Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Stochastic Grönwall Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 129 130

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Chapter 1

Introduction

This chapter introduces the brief, outlining its structure and motivating the core themes. We also establish and collect notation used throughout the book.

1.1 Motivation and Description of the Brief This brief comprises three chapters, increasing in complexity, described below: • In Chap. 2 we present a “classical” construction of the Itô stochastic integral, for processes evolving in a Hilbert space. This is introduced first for a one dimensional driving Brownian motion, before generalizations to other one dimensional martingales and, further, to cylindrical Brownian motion. Our construction is direct and designed to be familiar to a reader who has undertaken the real valued study as covered, for example, in [44, 55]. In defining the infinite dimensional Brownian motion, that is the cylindrical Brownian motion, we cover the fundamentals of martingale theory in Hilbert spaces broadly by finite dimensional projections along with the real valued theory. The hope is again that this approach is entirely accessible to a reader with background in the real valued integration theory. Precise attention must be paid to the martingale theory in order to properly consider Stratonovich equations; in our opinion the most thorough presentation of this material is in [60], yet more details and results are needed, such as the cross-variation between a Hilbert space valued and real valued martingale. The cylindrical Brownian motion is the only infinite dimensional driving process that we integrate with respect to; while we present a background on general Q-cylindrical processes which could be viable integrators, limiting ourselves to cylindrical Brownian motion enables the integral to be established as a straightforward limit of the integrals against finite dimensional Brownian motions. In particular we avoid the operator theoretic technicalities necessary in the general case, present in the all of the constructions of the classical works © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 D. Goodair, D. Crisan, Stochastic Calculus in Infinite Dimensions and SPDEs, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-69586-5_1

1

2

1 Introduction

[20, 38, 51, 52, 57, 60]. We hope that removing some generality makes our approach more accessible for newcomers to the field, and we note that our construction is sufficient for the framework and applications that follow. • Chapter 3 details a framework for the study of stochastic partial differential equations (SPDEs), which are evolution equations involving integrals of the form introduced in the previous chapter. Through this framework we define notions of solutions for an abstract SPDE, motivated in particular by the recent attention given to transport type noise (where the stochastic integral is dependent on the gradient of the solution) and Stratonovich equations. Motivation for such study is given below this list of contents. A rigorous mathematical understanding of these equations presents difficulty for two key reasons. The first is the gradient dependency in the noise, taking us beyond the most general “variational frameworks” seen in the literature as these are posed for a noise operator which is bounded on some Hilbert space. The second is the Stratonovich integration, which we are likely to only understand as a corrected Itô integral, yet this conversion is highly nontrivial for a noise which is not bounded on a Hilbert space. Furthermore we wish to consider nonlinear SPDEs, such as the evolution equations of fluid dynamics, rendering the well-established linear theory insufficient. To be precise, we present a framework which shares its spirit with the variational approach to SPDEs pioneered by Pardoux in the 1970s and now best represented in the more recent books [51, 56, 57]. This classical framework considers an evolution equation with respect to a Gelfand Triple, say .V ͨ− → H ͨ− → V ∗ , where solutions have paths which are square integrable in V , continuous in H , and satisfying an identity in .V ∗ . Recalling our motivation of fluid equations, the prototypical example in this framework is the Navier– Stokes equation. While analytically weak solutions fit this framework seamlessly, analytically strong solutions fit to the spaces .W 2,2 ͨ− → W 1,2 ͨ− → L2 , which prompts our choice of a triplet of embedded Hilbert spaces without any necessary duality structure. Furthermore, the aforementioned works allow only for a noise operator bounded in H (thus not of first order in applications) and do not consider Stratonovich equations. To include a Stratonovich transport type noise, we introduce a fourth Hilbert space, necessary as the Itô–Stratonovich correction requires an additional derivative to cover the transport type noise. We can then properly define weak, strong, and local solutions of nonlinear PDEs with Stratonovich transport noise, alongside other more classical additive and multiplicative stochastic perturbations. We believe that presenting the technical details, in such generality, of these notions here facilitates the rigorous and free analysis of the equations in future works. • Chapter 4 contains advanced novel techniques in the existence theory for nonlinear SPDEs. The beauty of the classical variational approach comes from the existence results, which certainly cannot be matched as elegantly in a framework built for 3D Navier–Stokes equations and related stochastic fluid models. Instead we focus on techniques that can be used in this direction, centered around the Galerkin Method in which finite dimensional approximations of the SPDE are

1.1 Motivation and Description of the Brief

3

considered and some properties are used to deduce their limit. Immediately then an existence result for the finite dimensional equations is required, more precisely for where the Hilbert space in which the equation evolves is finite dimensional, but the driving Brownian motion is still infinite dimensional. We assume standard Lipschitz and linear growth conditions, and to the best of our knowledge this result is not present in the literature. There are two predominant ways to deduce the existence of a limit of the finite dimensional approximations, which we detail now. The first is through tightness, which is the stochastic route to relative compactness arguments used in PDE theory. The idea is that from tightness we can deduce relative compactness of the laws of the processes over some suitable function space, at which point Skorohod’s Representation Theorem enables the deduction of a limiting process almost surely on a new probability space. Criteria to deduce tightness in relevant function spaces are thus of great significance, and our criteria come largely from the works of [3, 43, 59]. The second is through a Cauchy type argument in the relevant spaces, difficult to execute in the case of local solutions, but recently this has been overcome to great effect due to GlattHoltz and Ziane in [33] and extended by the authors here. We defer a greater discussion of this highly technical result to Chap. 4 and emphasize that this is a new result in the cutting edge theory of SPDEs. An energy equality in this setting is also presented. This is well understood in the typical variational framework, for which we again refer to [51, 57], but we take care in addressing some subtle differences. The first is the loss of the duality structure, though for this result we do assume a bilinear form relation which behaves similarly. The second is that we conduct the proof for local solutions, necessary for our motivating class of equations, so it is important for us to explicitly address how the localization affects the proof. Indeed, the consideration of local solutions, as well as the related localization in the construction of the integral, martingale theory, and analytical techniques, is an important extension of the framework of [51, 56, 57]. Similarly, the final key change is that we do not assume any integrability over the probability space of our processes, demanding again another source of localization which we find worthy of detailing in this brief. The chapter rounds out with a demonstration that the infinite dimensional noise can be reduced to one dimensional objects if it is constant multiplicative in each direction. Before setting up notation and beginning with our exposition, we give some more explicit motivation for this brief. First of all, why work in infinite dimensions? Finite dimensional stochastic differential equations have rich applications in physics and finance, for example, in Langevin equations modeling the movement of a particle in space [13] or the Black–Scholes options pricing model for the dynamics of the price of a stock [8]. These are applications of classical Itô calculus, where the integral of a process takes values in Euclidean spaces. While this theory is adequate in such applications, mathematical models for physical phenomena far exceed those for the position of a particle or that of a tradable stock price. The extension to infinite

4

1 Introduction

dimensions is necessary for stochastic differential equations modeling functions of both space and time, such as the velocity or temperature of a fluid. It is therefore necessary to define the stochastic integral ⎰

t

Ψ s dWs ,

.

(1.1)

0

for a class of stochastic processes .Ψ : Ω × [0, ∞) × Rn → Rd . We regard .Ψ not as a pointwise defined function but rather an element of a function space, which is our motivating context for stochastic integration of Hilbert space valued processes. The recent attention toward Stratonovich and transport noise SPDEs is inspired from the seminal work [42], in which Holm establishes a new class of stochastic equations which serve as fluid dynamics models by adding uncertainty in the transport of fluid parcels to reflect the unresolved scales. For recent literature on the analysis of equations under this stochastic scheme, please see [4, 11, 15– 19, 22, 34, 35, 37, 40, 41, 47, 50, 61] to list only a few, all of which rely on an Itô–Stratonovich conversion and a framework such as we present here, which is yet to see any rigorous justification. In fact the pertinence of Stratonovich transport noise in fluid dynamics equations was demonstrated as early as 1992 in the paper [9], and the analysis of SPDEs with general Stratonovich transport noise can be seen in the papers [1, 2, 5, 6, 12, 23–25, 27–31, 39, 48, 49, 53, 54] as well as the recent book [26]. We believe that our framework and results facilitate the rigorous and free analysis of such equations in future works.

1.2 Notation Throughout the brief, we work with a fixed filtered probability space .(Ω, F, (Ft ), P), which is complete with respect to .F0 . We always consider Banach spaces as measure spaces equipped with the corresponding Borel .σ -algebra and shall use .λ to denote the Lebesgue Measure. All of our Hilbert Spaces are assumed to be separable. Notation 1.1 Let .(X, μ) denote a general measure space, .(Y, ‖·‖Y ) and .(Z, ‖·‖Z ) be Banach Spaces, and .(U, 〈·, ·〉U ), .(H, 〈·, ·〉H ) be general Hilbert spaces: • .Lp (X; Y) is the usual class of measurable p-integrable functions from .X into .Y, .1 ≤ p < ∞, which is a Banach space with norm p . ‖φ‖ p L (X;Y)

⎰ :=

X

p

‖φ(x)‖Y μ(dx).

The space .L2 (X; Y) is a Hilbert Space when .Y itself is Hilbert, with the standard inner product

1.2 Notation

5

⎰ .

〈φ, ψ〉L2 (X;Y) =

X

〈φ(x), ψ(x)〉Y μ(dx).

• .L∞ (X; Y) is the usual class of measurable functions from .X into .Y which are essentially bounded, which is a Banach Space when equipped with the norm .

‖φ‖L∞ (X;Y) := inf{C ≥ 0 : ‖φ(x)‖Y ≤ C for μ-a.e. x ∈ X}.

• .C(X; Y) is the space of continuous functions from .X into .Y. • .Cw (X; Y) is the space of “weakly continuous” functions from .X into .Y, by which we mean continuous with respect to the given topology on .X and the weak topology on .Y. • .L (Y; Z) is the space of bounded linear operators from .Y to .Z. This is a Banach Space when equipped with the norm .

‖F ‖L (Y;Z) = sup ‖F y‖Z . ‖y‖Y =1

L (Y; Z) is the dual space .Y∗ when .Z = R, with operator norm .‖·‖Y∗ . • .L 1 (U; H) is the space of trace-class operators from .U to .H, defined as the elements .F ∈ L (U; H) such that for some basis .(ei ) of .U, .

∞ ⎲ .

‖F ei ‖H < ∞.

i=1

This is independent of the choice of basis (see, e.g., [14, pp. 267 Ex 20]). • .L 2 (U; H) is the space of Hilbert–Schmidt operators from .U to .H, defined as the elements .F ∈ L (U; H) such that for some basis .(ei ) of .U, ∞ ⎲ .

‖F ei ‖2H < ∞.

i=1

This is a Hilbert space with inner product

.

〈F, G〉L 2 (U;H) =

∞ ⎲

〈F ei , Gei 〉H ,

i=1

which is independent of the choice of basis (see, e.g., [14, pp. 267 Ex 20]). • For any .T > 0, .ST is the subspace of .C ([0, T ]; [0, T ]) of strictly increasing functions. • For any .T > 0, .D ([0, T ]; Y) is the space of cádlág functions from .[0, T ] into .Y. It is a complete separable metric space when equipped with the metric

6

1 Introduction

⎾ d(φ, ψ) := inf

.

η∈ST

⏋ sup |η(t) − t| ∨ sup ‖φ(t) − ψ(η(t))‖Y ,

t∈[0,T ]

t∈[0,T ]

which induces the so-called Skorohod topology (see [7, pp 124] for details). • The total variation of a function .φ : [0, T ] → Y, .VYT (φ), is defined as VYT (φ) := sup

k−1 ⎲

.

I

‖φ(ti+1 ) − φ(ti )‖Y

i=0

for the supremum taken over all partitions .I = {0 = t0 < t1 < · · · < tk = T }. A function .ψ : [0, ∞) → Y is said to be of bounded-variation if .VYT (ψ) < ∞ for all .T ≥ 0. For the reader’s convenience we also collect notation that is introduced throughout the brief, referencing where it is defined as relevant: 1A denotes the indicator function of the set A. IH T is defined in Definition 2.4. H .I is defined in Definition 2.5. 2 2 .M , Mc are defined in Definition 2.7. 2 .Mc (H) is defined in Definition 2.14. ¯ 2, M ¯ 2 (H) are the corresponding semimartingale spaces. • .M c c H • .IH,T , I M are defined in Definition 2.8. M H,T ¯ H ¯ • .IN , IN are defined in Definition 2.9. • .[·] is defined in Definition 2.16. • .[·, ·] is defined in Definition 2.18. • .IH T (W) is defined in Definition 2.19. • .IH (W) is defined in Definition 2.19. • .IH T (W) is defined in Definition 2.21. • .IH (W) is defined in Definition 2.21. • • • • •

. .

Chapter 2

Stochastic Calculus in Infinite Dimensions

In this chapter we present a “classical” construction of the Itô stochastic integral, for processes evolving in a Hilbert space. This is introduced first for a one dimensional driving Brownian motion, before generalizations to other one dimensional martingales and, further, to cylindrical Brownian motion. The construction is direct and designed to be familiar to a reader who has undertaken the study of integration with respect to a real valued Brownian motion. In addition, we offer a thorough introduction to martingale theory in Hilbert spaces.

2.1 A Classical Construction for Hilbert Space Valued Processes The construction of the stochastic integral for Hilbert space valued processes precisely mirrors the standard one dimensional Itô integral; as such we start from simple processes. .H will denote a general separable Hilbert space, with norm and inner product given by .‖·‖H and .〈·, ·〉H , respectively. W represents a standard one dimensional Brownian motion with respect to the fixed filtered probability space .(Ω; F, (Ft ), P). Definition 2.1 Let .0 = t0 < · · · < ti < ti+1 < . . . be a partition of the time interval .[0, ∞) such that .limi→∞ ti = ∞ approaches infinity. A simple .H-valued process is one that for .P − a.e. ω takes the form Ψ t (ω) = a0 (ω)1{0} (t) +

∞ ⎲

.

ai (ω)1(ti ,ti+1 ] (t),

i=0

( ) where each .ai ∈ L2 Ω; H and is .Fti -measurable, with respect to the Borel sigma algebra on .H. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 D. Goodair, D. Crisan, Stochastic Calculus in Infinite Dimensions and SPDEs, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-69586-5_2

7

8

2 Stochastic Calculus in Infinite Dimensions

Definition 2.2 The Itô integral of a simple .H-valued process .Ψ, with respect to Brownian motion, is defined as ⎰ .

t

Ψ s dWs :=

0

∞ ⎲ ( ) ai Wti+1 ∧t − Wti ∧t . i=0

Note that in reality the above is a finite sum, so the right hand side is well defined. Indeed, it can alternatively be expressed as k−1 ⎲ ( ) ( ) . ai Wti+1 − Wti + ak Wt − Wtk ,

(2.1)

i=0

where k is such that .t ∈ (tk , tk+1 ]. We define the integral for a more general class of integrands, using approximations by simple processes. For this we introduce the notion of progressive measurability. Definition 2.3 An .H-valued process .Ψ is said to be progressively measurable if for every .T > 0, the restricted process .Ψ : Ω × [0, T ] → H is measurable with respect to the product measure .FT × B ([0, T ]). Definition 2.4 We use .IH T to denote the class of .H-valued processes .Ψ which are progressively measurable and satisfy the square integrability condition ⎛⎰

T

E

.

0

⎞ ‖Ψ s ‖2H ds

< ∞.

(2.2)

( ) In other words, .Ψ ∈ L2 Ω × [0, T ]; H , where the domain space .Ω × [0, T ] is a measure space equipped with the product measure .P × λ.1 We have given the definition for progressively measurable processes, not previsible processes2 as will commonly be seen in the literature. Progressive measurability is a weaker condition than previsibility, but thankfully most reasonably behaved processes (adapted and left continuous for example) are both progressively measurable and previsible. We make the definition here for the more general class of integrands in the cases where the integrator is continuous. Other authors (e.g., [46]) may opt for previsible processes as these become necessary in retaining nice properties (e.g., martingality) when defining the stochastic integral with respect to discontinuous integrators.

1 The

progressive measurability of .Ψ ensures that it is measurable over this product space, and Tonelli’s theorem justifies exchanging the order of integration. 2 An .H-valued process is said to be previsible if it is measurable with respect to the sigma algebra generated by all left continuous adapted processes.

2.1 A Classical Construction for Hilbert Space Valued Processes

9

Definition 2.5 The class of processes .Ψ such that .Ψ ∈ IH T for all .T > 0 is denoted H by .I . IH represents our class of integrands for all times, though there will be nothing wrong with defining the integral for times .t ≤ T in the class .IH T . The construction comes as a limit of simple integrals, for which we need the following proposition which holds no differently to [55, Lemma 3.1.5] for example. .

n Proposition 2.1 For any .Ψ( ∈ IH T , there exists ) a sequence of simple processes .(Ψ ) 2 which converge to .Ψ in .L Ω × [0, T ]; H .

This approximation is the final piece of the constructive jigsaw, allowing us to define the stochastic integral below. Definition 2.6 We define the Itô stochastic integral for processes .Ψ ∈ IH by ⎰ .

t

⎰ Ψ s dWs := lim

t

n→∞ 0

0

Ψ ns dWs ,

(2.3)

where .(Ψ n ) is the (sequence of simple processes postulated in Proposition 2.1 which ) ( ) approach .Ψ in .L2 Ω × [0, t]; H , and the limit is taken in .L2 Ω; H . The fact that this is the natural topology in which to take the limit of simple stochastic integrals falls from the Itô isometry for simple processes, which further justifies that the construction is independent of the choice of simple approximation. Proposition 2.2 For a simple process .Ψ n and any time .t > 0, ⎛|| ⎰ t ||2 ⎞ ⎛⎰ t ‖ ‖ ⎞ || || ‖Ψ n ‖2 ds . =E E || Ψ ns dWs || s H

.

H

0

(2.4)

0

Proof Let us suppose that .Ψ n takes the form Ψ nt (ω) = a0n (ω)1{0} (t) +

∞ ⎲

.

n ] (t) ain (ω)1(tin ,ti+1

(2.5)

i=0

as outlined in Definition 2.1. Then applying Definition 2.2, we deconstruct the LHS of (2.4): ⎛|| ⎲ ⎞ ⎛||⎰ t ∞ ||2 ⎞ ( )||||2 || || || n ∧t − Wt n ∧t || = E || E || Ψ ns dWs || ain Wti+1 i

.

0

H

H

i=0

=E

⎛/ ⎲ ∞

ain

⎞ ∞ ( ) ⎲ ( )\ n n n n n aj Wtj +1 ∧t − Wtj ∧t Wti+1 ∧t −Wti ∧t , j =0

i=0

=

H

∞ ∞ ⎲ \ ⎲ (/ ) n ∧t − Wt n ∧t )(Wt n ∧t − Wt n ∧t ) E ain , ajn (Wti+1 i j j +1 i=0 j =0

H

10

2 Stochastic Calculus in Infinite Dimensions

recalling once more that the infinite sum is actually a finite sum (2.1) so there is no difficulty in extracting it from the inner product and expectation. For .i /= j , and without loss of generality .i < j , the random inner product is .Ftj measurable / as the \ continuity of the inner product preserves measurability, and n ∧t − Wt n ∧t ) and .(Wt n ∧t − Wt n ∧t ) are independent from therefore . ain , ajn (Wti+1 i j j +1 H the independent increments of Brownian motion. The terms thus vanish, and we are left with ⎞ ⎛ ∞ ⎲ ‖ n ‖2 2 ‖ ‖ n n . E ai H (Wti+1 ∧t − Wti ∧t ) i=0

to which we note independence again and assert that this is just

.

∞ ⎲ (‖ ‖2 )( n ) E ‖ain ‖H ti+1 ∧ t − tin ∧ t , i=0

which is precisely the integral ⎰ t⎲ ∞ (‖ ‖2 ) n ] (s)ds. . E ‖ain ‖H 1(tin ,ti+1 0 i=0

We can write ‖ ‖2 ‖ n ‖2 ‖ n ‖a ‖ 1(t n ,t n ] (s) = ‖ n ,t n ] (s)‖ , a 1 ‖ (t i H i i i+1 i i+1

.

H

the infinite sum of which is a single nonzero term, equal to ‖ ‖2 ∞ ‖ ‖ ⎲ ‖ n ‖ n n ] (s)‖ .‖a0 (ω)1{0} (s) + ai (ω)1(tin ,ti+1 ‖ ‖ i=0

H

at every s except for zero which is a set of Lebesgue measure zero in .[0, t]. Again the infinite sum is truly a single nonzero term, justifying its exchange with expectation, ‖ ‖2 ⨆ ⨅ and the above is of course .‖Ψ ns ‖H , proving the result. So, why is this useful in terms of the limit in (2.3)?( First) and foremost, it ensures that the limit is uniquely defined; given that .L2 Ω; H is complete, we need only to show that the sequence integrals is(Cauchy ( ⎰ t of stochastic ) ) in this space. The Itô isometry tells us that . 0 Ψ ns dWs is Cauchy in .L2 Ω; H if and only if ( ) the sequence .(Ψ n ) is Cauchy in .L2 Ω × [0, t]; H , which is of course true as by definition the .(Ψ n ) are convergent (to .Ψ) in this space. Furthermore, the isometry extends to the general integral defined in Definition 2.6, as a trivial corollary of the discussion here.

2.1 A Classical Construction for Hilbert Space Valued Processes

11

Corollary 2.1 The Itô isometry (2.4) holds for all processes .Ψ ∈ IH . Without direct appeal to the formal construction, we may also understand the integral (1.1) as a random element of the dual space .H∗ and identify the functional with its counterpart in .H in the usual sense. Theorem 2.1 The Itô stochastic integral defined in Definition 2.6 is the unique element of .H satisfying the duality relation /⎰

⎰

\

t

Ψ s dWs , φ

.

0

H

=

t

0

〈Ψ s , φ〉H dWs

(2.6)

for all .φ ∈ H. The above are random inner products, defined by 〈Ψ s , φ〉H (ω) := 〈Ψ s (ω), φ〉H

.

and similarly for the LHS. Remark 2.1 As a corollary, by the Riesz Representation Theorem, it is consistent to define the Itô stochastic integral as an .H∗ -valued random variable via the mapping ⎛⎰ φ |→

.

0

t

⎞ 〈Ψ s , φ〉H dWs (ω).

Proof Given that we have defined (1.1) as a limit of simple processes, it will come as no surprise that we use this approach to prove the relation (2.6). We will prove( that the result) holds for simple processes .Ψ n and later that it is preserved in the .L2 Ω × [0, t]; H limit. First though we ought to verify that the RHS of (2.6) makes sense, that is to say .〈Ψ, φ〉H is a valid (one dimensional) integrand. Thus we must show the standard progressive measurability and square integrability conditions: For the former, note that the progressive measurability of .Ψ is preserved under composition with the continuous mapping .〈·, φ〉H . The latter is straightforward, as ⎛⎰ E

.

0

T

⎞ ⎛⎰ 〈Ψ s , φ〉2H ds ≤ E

T

0

⎞ ⎛⎰ ‖Ψ s ‖2H ‖φ‖2H ds = ‖φ‖2H E

T 0

⎞ ‖Ψ s ‖2H ds ,

(2.7) which is finite by (2.2). Let us suppose that .Ψ n takes the form (2.5). Then applying Definition 2.2, we deconstruct the LHS of (2.6): /⎰

t

.

0

Ψ ns dWs , φ

\ H

=

∞ /⎲

( ) \ n ∧t − Wt n ∧t , φ ain Wti+1 i

i=0

=

∞ / ⎲ ( ) \ n ∧t − Wt n ∧t , φ ain Wti+1 i i=0

=

∞ ⎲ 〈 i=0

ain , φ

H

H

) 〉 ( n ∧t − Wt n ∧t Wti+1 H i

12

2 Stochastic Calculus in Infinite Dimensions

and proceed similarly for the RHS, observing that the integrand ∞ / ( ( ) ⎲ ) \ n ] (s) , φ 〈Ψ s , φ〉H = a0n 1{0} (s) + ain 1(tin ,ti+1

.

i=0

H

∞ / ⎲ ( 〈 ( ) 〉 ) \ n ] (s) , φ ain 1(tin ,ti+1 = a0n 1{0} (s) , φ H + i=0

H

∞ 〈 n 〉 ( 〈 〉 ( ) ⎲ ) n ] (s) ai , φ H 1(tin ,ti+1 = a0n , φ H 1{0} (s) + i=0

is again simple. This is completely analogous to showing that .〈Ψ, φ〉H was a valid integrand. Applying Definition 2.2 to the above proves the result for simple processes, so all that remains to show is preservation in the limit. By definition /⎰

\

t

Ψ s dWs , φ

.

0

H

=

⎰

/

t

lim

n→∞ 0

Ψ ns dWs , φ

\ H

( ) and a reminder that this limit is taken in .L2 Ω; H . We would like to take the limit outside of the inner product, in some appropriate topology, and use the result for simple functions: The steps would be ⎰

/ .

lim

n→∞ 0

t

Ψ ns dWs , φ

\ H

= lim

/⎰

n→∞

⎰ = lim

n→∞ 0

t

0 t〈

Ψ ns dWs , φ

Ψ ns , φ

〉 H

\ H

dWs

so (it should ) be clear that the topology we want to take this limit in is that of L2 Ω; R , as the last line would be precisely the RHS of (2.6) by definition if n we can ( show that the ) simple real valued process .〈Ψ , φ〉H converges to .〈Ψ, φ〉H 2 in .L Ω × [0, t]; R . Thankfully, it is straightforward to justify taking this limit ( ) outside of the inner product: If .(fn ) converges to f in .L2 Ω; H , then

.

⎞2 ⎞2 ⎞ ⎛ ⎛ ⎛ E 〈fn , φ〉H − 〈f, φ〉H = E 〈fn − f, φ〉H ≤ E ‖fn − f ‖2H ‖φ‖2H ⎞ ⎛ = ‖φ‖2H E ‖fn − f ‖2H −→ 0

.

( ) so .(〈fn , φ〉H ) converges to .〈f, φ〉H in .L2 Ω; R , as required to justify the inter( ) change. To show the convergence of .〈Ψ n , φ〉H to .〈Ψ, φ〉H in .L2 Ω × [0, t]; R , we apply the same argument:

2.1 A Classical Construction for Hilbert Space Valued Processes

‖ 〈 〉 ‖ ‖〈Ψ, φ〉H − Ψ n , φ ‖ (

.

H L2 Ω×[0,t];R

13

‖ ‖ ) = ‖〈Ψ − Ψ n , φ 〉 ‖ ( ) H L2 Ω×[0,t];R =E

⎛⎰ t 〈 0

Ψ s − Ψ ns , φ

〉2

⎞

ds H

⎞ ⎛⎰ t ‖ ‖ ‖Ψ s − Ψ n ‖2 ‖φ‖2 ds ≤E s H H 0

= ‖φ‖2H E =

⎛⎰ t ‖ ⎞ ‖ ‖Ψ s − Ψ n ‖2 ds s H

‖

‖φ‖2H ‖Ψ

0

‖ − Ψ n ‖L2 (Ω×[0,t];H)

−→ 0, where convergence to 0 is by definition of the approximating sequence .Ψ n .

⨆ ⨅

We provide two applications of this result below, both of which will be fundamental to our SPDE framework. Proposition 2.3 The Itô isometry holds for a multidimensional driving Brownian motion, in the sense that if .(Ψ i )ni=1 are a collection of processes in .IH , and .(W i )ni=1 are independent Brownian motions, then ⎛|| ⎲ n ⎰ t n ||2 ⎞ ⎲ ⎛ ⎰ t ‖ ‖2 ⎞ || ‖ i‖ || = E || Ψ is dWsi || E ‖Ψ s ‖ ds .

.

i=1

H

0

H

0

i=1

Proof We look to simplify the left hand side of the required equality, applying Parseval’s identity (e.g., [45, pp. 170]) for a basis .(ek ) of .H: ⎛|| ⎲ n ⎰ t ∞ /⎲ n ⎰ t ||2 ⎞ \2 ⎲ || || =E E || Ψ is dWsi || Ψ is dWsi , ek

.

i=1

H

0

k=1

=E

H

0

i=1

∞ ⎛⎲ n ⎰ t/ ⎲ k=1

i=1

0

Ψ is , ek

\ H

dWsi

⎞2

having used linearity of the inner product to pull out the sum and Theorem 2.1. We can now regard the infinite sum as an integral with respect to the counting measure and apply Tonelli’s theorem, also expanding the square to obtain

.

∞ n ⎰ t/ \ ⎞2 ⎛⎲ ⎲ Ψ is , ek dWsi E k=1

i=1

H

0

=

∞ ⎲ k=1

E

n ⎛⎰ t / n ⎲ ⎲ i=1 j =1

0

Ψ is , ek

\ H

dWsi

⎞⎛ ⎰ t / \ ⎞ j j Ψ s , ek dWs . 0

H

14

2 Stochastic Calculus in Infinite Dimensions

For the cross terms .i /= j we make use of the independence of the Brownian motions and hence the respective stochastic integrals, as well as the standard property that the Itô integral has zero expectation to nullify these terms. Our expression reduces to .

n ∞ ⎲ \ ⎞2 ⎛⎰ t / ⎲ Ψ is , ek dWsi E H

0

k=1 i=1

to which we can apply the Itô isometry (Corollary 2.1 for the Hilbert space .R, which is of course the standard isometry) giving us n ∞ ⎲ ⎲ .

k=1 i=1

⎰ t/ E 0

Ψ is , ek

\2 H

ds.

From this, we apply Tonelli twice more to take the infinite sum all the way through, n ⎲ .

i=1

E

⎰ t⎲ ∞ / 0 k=1

Ψ is , ek

\2 H

ds.

A final application of Parseval’s identity gives the result.

⨆ ⨅

While we chose to prove Proposition 2.2 and subsequently Corollary 2.1 from first principles in the Hilbert space setting, the method of proof here touches upon a fundamental aspect of this theory: With a good understanding of the standard setting in .R, we can apply Theorem 2.1 to straightforwardly deduce key properties here. Indeed, if we accepted the Itô isometry in .R, we could have just proven Corollary 2.1 in the simple vein of Proposition 2.3. We take this approach in extending the result of Theorem 2.1. Theorem 2.2 Suppose that .H1 , H2 are Hilbert spaces such that .Ψ ∈ IH1 and .T ∈ L (H1 ; H2 ). Then the process .T Ψ defined by ( ) (T Ψ)s (ω) = T Ψ s (ω)

.

belongs to .IH2 and is such that ⎛⎰ T

.

0

t

⎞ ⎰ t Ψ s dWs = T Ψ s dWs .

(2.8)

0

Proof We shall prove first that .T Ψ ∈ IH2 . The progressive measurability is preserved under the continuity of T , and for C the (square of the) boundedness constant associated to T , we have that at any time t,

2.2 Martingale and Local Martingale Integrators

⎛⎰ E

.

0

t

‖T Ψ s ‖2H2 ds

⎞

15

≤ CE

⎛⎰

t

0

⎞ ‖Ψ s ‖2H1 ds < ∞

as .Ψ ∈ IH1 , showing that .T Ψ ∈ IH2 . To commute T with the integral, we shall use the characterization from Theorem 2.1, having now established that the right hand side of (2.8) is well defined in .H2 . We introduce .T ∗ ∈ L (H2 ; H1 ) as the adjoint of T and observe that for any .φ ∈ H2 , / ⎛⎰ t ⎞ \ T Ψ s dWs , φ

.

0

H2

=

/⎰

Ψ s , T ∗φ

= 0

t

= 0

=

Ψ s dWs , T ∗ φ

0 t〈

⎰ ⎰

t

/⎰

〉 H1

\ H1

dWs

〈T Ψ s , φ〉H2 dWs \

t

T Ψ s dWs , φ 0

H2

applying Theorem 2.1 twice. As this equality holds for arbitrary .φ ∈ H2 , then we have proven (2.8), which is of course an identity .P − a.e. in .H2 . ⨆ ⨅

2.2 Martingale and Local Martingale Integrators As expected, we can extend the definition to integrators beyond Brownian motion, in the same manner as the standard Itô integral. We begin the extension to continuous square integrable martingales and then to continuous local martingales. Definition( 2.7 We ) shall denote the class of real valued martingales M such that Mt ∈ L2 Ω; R for every .t ≥ 0 by .M2 . The subclass of such martingales with 2 .P − a.s. continuous paths will be represented by .Mc . .

Definition 2.8 For any .M ∈ M2c , define .IH,T to be the class of .H-valued M processes .Ψ which are progressively measurable on .[0, T ]×Ω and satisfy the square integrability condition ⎛⎰

T

E

.

0

⎞ ‖Ψ s ‖2H d[M]s < ∞,

where .[M]· is the quadratic variation of .M· . We similarly denote by .IH M the class of H,T processes .Ψ such that .Ψ ∈ IM for all .T > 0.

16

2 Stochastic Calculus in Infinite Dimensions

Constructing the integral ⎰

t

Ψ s dMs

.

0

for .Ψ ∈ IH M now falls from what we have already done for (1.1). We use simple processes .Ψ n as in Definition 2.1 to approximate .Ψ, in the sense that ⎛⎰ .

⎞ ‖ ‖ ‖Ψ s − Ψ n ‖2 d[M]s = 0. s H

T

lim E

n→0

0

Simply replacing W by M in Definitions 2.2 and 2.6 completes the construction, so we do not give the details here. Let us now move on to the more delicate matter of integration with respect to a local martingale. This begins again with notation for our set of integrands. H,T

Definition 2.9 For a continuous local martingale N, define .I¯ N progressively measurable processes .Ψ such that ⎰ .

0

T

to be the class of

‖Ψ s ‖2H d[N]s < ∞ P − a.e.

H

H,T

Also define .I¯ N to be those processes .Ψ in .I¯ N

(2.9)

for every T .

Suppose that N is localized by the stopping times .(Tn ). Without loss of generality, this sequence of stopping times can be chosen such that the stopped processes .N Tn defined by NtTn := Nt∧Tn

.

are bounded; if .(Tn' ) are localizing stopping times, then we can simply set Tn = Tn' ∧ inf{0 ≤ t < ∞ : |Nt | ≥ n}

.

so that for each n, .N Tn is a bounded continuous martingale and hence in .M2c . Note of course that the new stopping times .(Tn ) are still nondecreasing and approach infinity .P − a.s. by the pathwise continuity of N. Continuing in this manner, for a H process .Ψ ∈ I¯ N , let us define some more nondecreasing random times .(Rn ) by ⎰ Rn := n ∧ inf{0 ≤ t < ∞ :

.

0

t

‖Ψ s ‖2H d[N]s ≥ n}

(2.10)

taking the convention that the infimum of the empty set is infinite. The random variables .(Rn ) are stopping times as they are simply first hitting times of the continuous and adapted processes

2.2 Martingale and Local Martingale Integrators

⎰

t

.

0

17

‖Ψ s ‖2H d[N ]s .

Again these times tend to infinity .P − a.s. by condition (2.9). Now define .τn by τn = Rn ∧ Tn

(2.11)

.

and the truncated processes .Ψ n as Ψ nt := Ψ t 1t≤τn .

.

Understanding the fact that for .m ≤ n, and .t ≤ τm , we have .Ψ t 1t≤τn = Ψ t 1t≤τm and also that .Ntτn = Ntτm , and then eventually the sequence ⎛⎰ .

0

t

Ψ ns dNsτn

⎞

is constant .P − a.s.. Thus, we can make the following definition. Definition 2.10 In the setting described, we define ⎰

t

.

⎰ Ψ s dNs := lim

n→∞ 0

0

t

Ψ ns dNsτn

(2.12)

for the limit taken .P − a.s. in .H. Of course to do this we require that at any n, .Ψ n ∈ IH N τn : The process .(1t≤τn ) is progressively measurable, as it is both left continuous and adapted (adaptedness becomes clear when for each fixed t, we write the random variable .1t≤τn as .1 − 1t>τn ). The square integrability in Definition 2.8 comes from the fact that the random variable ⎰ t ‖ n ‖2 ‖Ψ ‖ d[N τn ]s . s H 0

is bounded by n .P − a.s. (owing to (2.10)), and hence, the expectation satisfies the same bound. Where N is itself a genuine martingale, this procedure defines the stochastic integral for processes with only the regularity (2.9). In this case we do not have to stop the integrator, and we just truncate the integrand. Definition 2.11 In the special case where the continuous local martingale is a H Brownian motion, we denote .I¯ W by simply .IH . This class of processes differs from H .I due to the assumption .P − a.s., i.e., (2.9), as opposed to (2.2) in expectation. We extend properties of the stochastic integral to this class of processes.

18

2 Stochastic Calculus in Infinite Dimensions H

Proposition 2.4 Let .Ψ ∈ I¯ and .φ ∈ L∞ (Ω; H) be .F0 -measurable. Then ¯ H , and for every .t > 0, we have that .〈Ψ, φ〉 ∈ I /⎰

\

t

Ψ r dWr , φ

.

0

⎰ H

t

= 0

〈Ψ r , φ〉H dWr

(2.13)

P − a.s.. The above are random inner products defined by

.

〈Ψ s , φ〉H (ω) := 〈Ψ s (ω), φ(ω)〉H

.

and similarly for the left hand side. R Proof We should first justify that .〈Ψ · , φ〉H ∈ I¯ . The progressive measurability follows as for every .T > 0 the mapping

〈Ψ · , φ〉H : t × ω × ω˜ |→ 〈Ψ t (ω), φ(ω)〉 ˜ H

.

is .B([0, T ]) × FT × F0 measurable, so in particular it is .B([0, T ]) × FT × FT measurable, and as such 〈Ψ · , φ〉H : t × ω |→ 〈Ψ t (ω), φ(ω)〉H

.

is .B([0, T ]) × FT measurable as required. Note that we have used the progressive measurability requirement on .Ψ. We also appreciate that for .P − a.e. .ω, ⎰

T

.

0

⎰ 〈Ψ r (ω), φ(ω)〉2H dr ≤ ‖φ(ω)‖2H

0

T

‖Ψ r (ω)‖2H dr < ∞

H R again by assumption on .Ψ ∈ I¯ . Thus .〈Ψ, φ〉H belongs to .I¯ . To compute the integrals we introduce the stopping times

⎧ ⎫ ⎰ t ‖Ψ r ‖2H dr ≥ j τj := j ∧ inf 0 ≤ t < ∞ : (1 + ‖φ‖2H )

.

0

〈 〉 such that for every .j ∈ N, .Ψ · 1·≤τj ∈ IH , . Ψ · 1·≤τj , φ H ∈ IR . It is sufficient to show that /⎰ t \ ⎰ t 〈 〉 Ψ r 1r≤τ j , φ H dWr . Ψ r 1r≤τ j dWr , φ = (2.14) 0

H

0

holds .P − a.e. for every j . We now fix an arbitrary .j ∈ N. The plan is as follows: We consider a sequence of simple processes .(Фn ) which approximate .Ψ · 1·≤τj in n 2 .L (Ω × [0, t]; H) as postulated in Proposition 2.1. We then claim that .(〈Ф , φ〉H ) 〈 〉 is a sequence of .R valued simple processes which converge to . Ψ · 1·≤τj , φ H in

2.2 Martingale and Local Martingale Integrators

19

L2 (Ω × [0, t]; R). Following this, we prove (2.13) for this simple case and show the identity holds in the limit. We first show that for each .n ∈ N, .〈Фn , φ〉H is a simple process. Let .Фn have representation as in Definition 2.1. Then

.

/ \ ∞ ⎲ 〈 n 〉 n n n ], φ . Ф ,φ = a0 1{0} + ai 1(tin ,ti+1 H H

i=0

〈 〉 = a0n , φ H 1{0} +

∞ ⎲

〈 n 〉 n ] ai , φ H 1(tin ,ti+1

i=0

so this would 〈 〉satisfy the requirements of an .R valued simple process if for each i ∈ N, . ain , φ H ∈ L2 (Ω; R) and is .Fti -measurable. For the square integrability constraint, observe that ⎛‖ ‖ ⎞ ⎛〈 ⎛‖ ‖ ⎞ 〉2 ⎞ 2 2 .E ain , φ H ≤ E ‖ain ‖H ‖φ‖2H ≤ ‖φ‖2L∞ (Ω;H) E ‖ain ‖H < ∞

.

by the assumptions of .ai ∈ L2 (Ω; H) and .φ ∈ L∞ (Ω; H). The .Fti measurability follows in the same way as the progressive measurability of .〈Ψ, φ〉H . Indeed the required .L2 (Ω × [0, t]; R) convergence follows similarly as ‖〈 〉 〈 〉 ‖ ‖ ‖ ‖ Ψ · 1·≤τj , φ H − Фn· , φ H ‖ 2 L (Ω×[0,t];R) ‖〈 〉 ‖ ‖ ‖ = ‖ Ψ · 1·≤τj − Фn , φ H ‖

.

L2 (Ω×[0,t];R)

‖‖ ‖ ‖ ‖ ‖ ≤ ‖‖Ψ · 1·≤τj − Фn ‖H ‖φ‖H ‖ 2 L (Ω×[0,t];R) ‖‖ ‖ ‖ ‖ ‖ ≤ ‖φ‖L∞ (Ω;H) ‖‖Ψ · 1·≤τj − Фn ‖H ‖ 2

L (Ω×[0,t];R)

,

and by assumption, ‖‖ ‖ ‖ ‖ ‖‖ ‖ Ψ · 1·≤τj − Фn ‖H ‖

.

L2 (Ω×[0,t];R)

−→ 0

as .n → ∞, so the convergence is proved. To show the identity (2.13) in the case of the simple process .Фn , observe that /⎰

t

.

0

\ Фnr dWr , φ

H

=

/∞ ⎲

ain

( ) n ∧t − Wt n ∧t , φ Wti+1 i

\ H

i=0

=

∞ / ⎲ i=0

⎛

⎞

n ∧t − Wt n ∧t , φ ain Wti+1 i

\ H

20

2 Stochastic Calculus in Infinite Dimensions

=

∞ ⎞ ⎲ 〈 n 〉 ⎛ n ∧t − Wt n ∧t ai , φ H Wti+1 i i=0

⎰

t

= 0

〈 n 〉 Фr , φ H dWr

as required. In order to conclude the argument, by definition of the integral, we have that ⎰ t ⎰ t . Ψ r 1r≤τ j dWr = lim Фnr dWr n→∞ 0

0

⎰

t

0

〈 〉 Ψ r 1r≤τ j , φ H dWr = lim

⎰

t

n→∞ 0

〈 n 〉 Фr , φ H dWr ,

where the first limit is taken in .L2 (Ω; H) and the second one in .L2 (Ω; R). For each we can thus extract a .P − a.s. convergent subsequence in the appropriate space, so by taking successive subsequences we can find one common subsequence indexed by .(nk ) such that the above limits hold .P − a.s.. Thus /⎰

\

t

.

0

Ψ r 1r≤τ j dWr , φ

/ H

=

⎰

t

lim

nk →∞ 0

/⎰

t

= lim

nk →∞

⎰ = lim =

nk →∞ 0 ⎰ t 〈 0

0 t

\ Фnr k dWr , φ \ Фnr k dWr , φ

H

H

〈 n 〉 Фr , φ H dWr

Ψ r 1r≤τ j , φ

〉 H

dWr

so (2.14) is justified and the proof is complete. ⨆ ⨅ We note that the .F0 -measurability requirement on .φ really comes into play in 〈 〉 showing the .Fti -measurability of . ain , φ H . If we were to consider the integral over some .[s, t] interval instead, then one could relax .φ to only being .Fs -measurable. In fact the result can also be extended to unbounded .φ. To do this we shall prove a Stochastic Dominated Convergence Theorem, Lemma 2.1. H Lemma 2.1 Let .(Ψ n ) be a sequence in .I¯ such that there exist processes .Ψ : Ω × [0, ∞) → H and .Ф ∈ L2 ([0, ∞); R) .P − a.s., with the properties that for every .T > 0, .P × λ − a.e. .(ω, t) ∈ Ω × [0, T ]: ‖ ‖ 1. .‖Ψ nt (ω)‖H ≤ |Фt (ω)| for all .n ∈ N. 2. .(Ψ nt (ω)) is convergent to .Ψ t (ω) in .H.

2.2 Martingale and Local Martingale Integrators

21

H Then .Ψ ∈ I¯ , and for every .t > 0, there exists a subsequence indexed by .(nk ) such that ⎰ t ⎰ t . lim Ψ nr k dWr = Ψ r dWr (2.15) nk →∞ 0

0

P − a.s..

.

Proof Immediately, we note that .Ψ inherits the progressive measurability from .Ψ n from the almost everywhere limit in the product space .Ω × [0, T ] when equipped with product sigma algebra .FT × B([0, T ]). Similarly we must have that for .P × λ − a.e. .(ω, t), .‖Ψ t (ω)‖H ≤ |Фt (ω)| so .Ψ ∈ L2 ([0, ∞); H) .P − a.s. hence H belongs to .I¯ . We look to find a common sequence of localizing times for the stochastic integrals and then demonstrate (2.15) by showing the identity holds true when stopped at each localizing time. To this end we introduce the stopping times ⎧ ⎫ ⎰ t 2 .τj := j ∧ inf 0 ≤ t < ∞ : |Фr | dr ≥ j , 0

which from the assumption 1 serve as a sequence of localizing times for every .Ψ n , and too for .Ψ. Thus for any fixed .t > 0 and .j ∈ N, we wish to show that ⎰ .

lim

nk →∞ 0

t

⎰ Ψ nr k 1r≤τj dWr

t

= 0

Ψ r 1r≤τj dWr

(2.16)

for a subsequence .(nk ) .P − a.s., or equivalently that ⎰ .

lim

nk →∞ 0

t

(Ψ nr k − Ψ r )1r≤τj dWr = 0.

We first assess the convergence in .L2 (Ω; H), applying Corollary 2.1 for each fixed n to see that ‖2 ‖⎰ t ⎛⎰ t ⎞ ‖ ‖ ‖ n ‖2 n ‖ ‖ ‖ ‖ (Ψ r − Ψ r )1r≤τj H dr . .E ‖ (Ψ r − Ψ r )1r≤τj dWr ‖ = E 0

H

0

Observing that for .P × λ − a.e. .(ω, t), ⎛‖ ‖ ‖ ‖ ⎞2 ‖ ‖ n ‖(Ψ (ω) − Ψ r (ω))1r≤τ (ω)‖2 ≤ ‖Ψ n (ω)1r≤τ (ω) ‖ + ‖Ψ r (ω)1r≤τ (ω)‖ r r j j j H H H

.

≤ 4|Фr (ω)1r≤τj (ω)|2 . Then with dominating function .4|Ф· 1·≤τj |2 , we can apply the standard Dominated Convergence Theorem for the integral over the product space (we face no problems

22

2 Stochastic Calculus in Infinite Dimensions

with the order of integration from Tonelli’s theorem given the progressive measurability) to deduce that ⎛⎰ .

t

lim E

n→∞

0

‖ n ‖ ‖(Ψ − Ψ r )1r≤τ ‖2 dr r j H

⎞ = 0,

and therefore ‖2 ‖⎰ t ‖ ‖ n ‖ . lim E (Ψ r − Ψ r )1r≤τj dWr ‖ ‖ = 0. ‖ n→∞ H

0

Thus we have justified the convergence (2.16) in the sense of .L2 (Ω; H), from which we can deduce a .P − a.s. convergent subsequence and the result is proved. ⨆ ⨅ H Proposition 2.5 Let .Ψ ∈ I¯ and .φ : Ω → H be .F0 -measurable. Then .〈Ψ, φ〉 ∈ H I¯ , and for every .t > 0 we have that

/⎰

\

t

Ψ r dWr , φ

.

⎰ H

0

t

= 0

〈Ψ r , φ〉H dWr

(2.17)

P − a.s..

.

R Proof A justification that .〈Ψ r , φ〉H ∈ I¯ is precisely as in Proposition 2.4. To apply this result, we rewrite .φ in a trivial way as

φ :=

∞ ⎲

.

φ1k≤‖φ‖H 0 that E

⎛/ ⎲ ∞ √

.

λi ei Wti , g

∞ \ /⎲ \ ⎞ √ j = 〈Qg, h〉H (t ∧ s). λj ei Ws , h H

i=1

H

j =1

We take the limit through the first inner product on the LHS as above, that is, ⎛⎛ E

.

n √ ∞ /⎲ \ ⎞/ ⎲ \ ⎞ √ j i , λi ei Wt , g λj ei Ws , h lim

n→∞

H

i=1

H

j =1

(2.21)

( ) and argue that for a sequence of functions .(fn ) convergent to f in .L2 Ω; R , and ( ) 2 .a ∈ L Ω; R , that (lim fn )(a) = lim(fn a).

.

L2

L1

The right side is well defined as the limit of a Cauchy sequence: ) ( ) ( ) ( ) ( E |fn a − fm a| = E |(fn − fm )a| ≤ E |fn − fm |2 E |a|2 ,

.

( ) and a similar calculation shows that this element of .L1 Ω; R is the left side. Applying to (2.21) and pulling the .L1 limit through the expectation produce

.

lim E

n→∞

⎛/ ⎲ n √ i=1

λi ei Wti , g

∞ \ /⎲ √ H

j =1

j

λj ej Ws , h

\ H

26

2 Stochastic Calculus in Infinite Dimensions

Playing the same game, this is ⎛/ ⎲ n √ m \ /⎲ \ ⎞ √ j i lim lim E λi ei Wt , g λj ej Ws , h

.

n→∞ m→∞

H

i=1

j =1

H

and further .

lim lim E

n→∞ m→∞

⎞ ⎛⎲ m √ n ⎲ 〈 〉 〉 〈√ j λi ei , g H λj ej , h H Wti Ws . i=1 j =1

Independence of the Brownian motions .(W i ) and the fact that .t ∧ s is the correlation function of Brownian motion gives that this is equal to

.

lim

n→∞

n √ ⎲ 〈

λi ei , g

〉 〈√ H

n ⎲ 〈 〉 〉 〈 〉 λi ei , g H ei , h H (t ∧ s), λi ei , h H (t ∧ s) = lim n→∞

i=1

i=1

which is just ∞ /⎲ .

λi 〈ei , g〉ei , h

\

i=1

.

H

(t ∧ s) = 〈Qg, h〉H (t ∧ s)

as required. =⇒ : For the reverse direction, assume that .WQ is regular with corresponding ~ Q (which is not necessarily of the form (2.19)). We want an expression process .W in terms of the trace of Q, so we exploit Definition 2.12:

.

∞ ∞ ⎲ ) ⎲ ( 2 〈Qei , ei 〉H t, E |WQ ei (t)| = i=1

i=1

and use an alternative expression from the assumed regular representation:

.

∞ ∞ \2 ) ⎲ ) ⎲ ( (/ Q 2 ~ (t), ei = E |WQ (t)| E W ei i=1

H

i=1

=E

∞ / ⎛⎲ i=1

~ Q (t), ei W

\2 ⎞ H

‖ (‖ ‖ ~ Q ‖2 ) = E ‖W (t)‖ < ∞, H

where the infinite sum is pulled inside the expectation from the Monotone Con( ) ~ Q ∈ L2 Ω; H . Hence vergence Theorem, and finiteness is by assumption on .W Q is trace-class, and by the first implication, (2.19) is a regular representation of .WQ . ⨆ ⨅

2.4 Martingale Theory in Hilbert Spaces

27

Remark 2.3 A standard cylindrical Brownian motion, that is, where Q is the identity, is thus not regular. In light of Remark 2.3, it would be convenient to have such a representation for cylindrical Brownian motion (denoted simply by .W); we would like this to be along the lines of Wt =

∞ ⎲

.

ei Wti ,

(2.22)

i=1

where the .(ei ) forms an orthonormal basis of .H, and .(W i ) are real valued independent Brownian motions. We can in fact explicitly construct a larger Hilbert space .H' such that the inclusion mapping .J : H ͨ− → H' is Hilbert–Schmidt. The ' ∗ composition .Q := J J is then trace-class on .H , and indeed .W is a Q-cylindrical Brownian motion on .H’: √ We defer the details to, e.g., [52, Problem 3.2.6]. To be precise, that is .J (ei ) = λi ηi for each i, where the .(ηi ) forms an orthonormal basis (of .H' ) of eigenfunctions of .J J ∗ with eigenvalues .λi , and for any .h ∈ H, Wh (t) =

/∞ ⎲

.

\ J (ei )Wti , J (h)

i=1

. H'

In this spirit, we consider (2.22) as a formal representation of cylindrical Brownian motion.

2.4 Martingale Theory in Hilbert Spaces We first define the notion of a Hilbert space valued martingale, before addressing martingale properties of the stochastic integral. Definition 2.14 A process M taking values in a Hilbert space .H is said to be a martingale if for every .h ∈ H, the process .〈M, h〉H is a real valued martingale. The martingale is said to be continuous if for .P − a.e. .ω and every .T > 0, .M(ω) : The martingale is said to be square integrable if for every [0, T ] →( H is continuous. ) 2 .t ≥ 0, .E ‖Mt ‖ The class of continuous square integrable martingales will < ∞. H be denoted .M2c (H).3 We look to show that .M2c (H) is closed in a suitable topology, for which we shall use a sufficient condition for continuity proven now.

3 Recall

that .M2c := M2c (R), see Definition 2.7.

28

2 Stochastic Calculus in Infinite Dimensions

Lemma 2.2 Let .ψ ∈ Cw ([0, T ]; H) and .‖ψ‖2H ∈ C ([0, T ]; R), that is, .ψ is weakly continuous with continuous norm. Then .ψ ∈ C ([0, T ]; H). Proof We fix some .t ∈ [0, T ] and look to show that .ψ is continuous at t. To this end consider arbitrary .s ∈ [0, T ]. Then ‖ψt − ψs ‖2H = 〈ψt − ψs , ψt − ψs 〉H

.

= 〈ψt − ψs , ψt 〉H − 〈ψt − ψs , ψs 〉H = 〈ψt − ψs , ψt 〉H + ‖ψs ‖2H − 〈ψt , ψs 〉H = 〈ψt − ψs , ψt 〉H + ‖ψs ‖2H + 〈ψt , ψt − ψs 〉H − ‖ψt ‖2H ⎞ ⎛ = 2〈ψt − ψs , ψt 〉H + ‖ψs ‖2H − ‖ψt ‖2H . For any given .ε > 0, there exists a .δ > 0 such that for all .s ∈ [0∨(t −δ), (t +δ)∧T ], ε , 3 ε ‖ψs ‖2H − ‖ψt ‖2H < 3 〈ψt − ψs , ψt 〉H
0, by Proposition A.2, we have that ⎫⎞ ⎛ ⎛⎧ ⎞ j j ⎨ ⎬ ⎲ ⎲ 1 〈Mt (ω), ei 〉2H > ε ⎠ ≤ E ⎝ 〈MT , ei 〉2H ⎠ , .P ⎝ ω ∈ Ω : sup ⎭ ⎩ ε t∈[0,T ]

.

i=k+1

i=k+1

(2.23) ) ( but we have already established that .E ‖MT ‖2H < ∞, or equivalently E

⎛∞ ⎲

.

⎞ 〈Mt (ω), ei 〉2H

< ∞.

i=1

Therefore in the limit as .k → ∞, (2.23) approaches zero uniformly in .j > k. One may choose a subsequence such that for all .kl < km ,

30

2 Stochastic Calculus in Infinite Dimensions

⎫⎞ ⎛⎧ km ⎨ ⎬ ⎲ 1 ⎠≤ 1. 〈Mt (ω), ei 〉2H > .P ⎝ ω ∈ Ω : sup ⎩ l⎭ 2l t∈[0,T ] i=kl +1

∑l 〈M, ei 〉2H is Cauchy in By the Borel Cantelli Lemma the subsequence . ki=1 .C ([0, T ]; R) .P − a.s.. It thus admits a limit .P − a.s. in .C ([0, T ]; R), which ∑ 2 2 agrees with the limit at each t, given by . ∞ i=1 〈Mt , ei 〉H , which is of course .‖Mt ‖H . 2 Therefore, the process .‖M‖H is pathwise continuous; from Lemma 2.2, it is now sufficient to just show weak continuity. This is clear however, as for any given .φ ∈ H and .t ≥ 0, we have again that .

⎛〈 〉2 ⎞ lim E Mtn − Mt , φ H = 0,

n→∞

∑ so .〈M, φ〉H is shown to belong to .M2c exactly as was done for . ki=1 〈M, ei 〉H , thus concluding the proof. ⨆ ⨅ Proposition 2.8 For a standard real valued Brownian motion W and .Ψ ∈ IH , the Itô stochastic integral ⎰

·

Ψ s dWs

.

0

belongs to .M2c (H). Proof Recalling Definition 2.6, the integral is defined at each time t as a limit in L2 (Ω; H) of simple integrals. From Proposition 2.7 it is sufficient to show that these approximating integrals all belong to .M2c (H), which is clear referring to the ⨆ ⨅ representation (2.1) and the definition of .M2c (H).

.

Proposition 2.8 extends to the case of a general martingale integrator as in the finite dimensional setting. Local martingality is then defined as we would expect, and we have the following result. H Proposition 2.9 For a continuous local martingale N and .Ψ ∈ I¯ N , the Itô stochastic integral

⎰

t

Ψ s dNs

.

0

is itself a continuous local martingale. Proof We claim that the localizing stopping times are given simply by the .(τn ) as defined in (2.11). We have already seen that these tend to infinity almost surely, so it just remains to show that at any fixed .n ∈ N the stopped process is a continuous ⨆ ⨅ martingale. The argument is now exactly as in Proposition 2.8.

2.4 Martingale Theory in Hilbert Spaces

31

The next ingredient would be a definition of quadratic variation, which we look to do via a Doob–Meyer decomposition for .M ∈ M2c (H) (Theorem A.1). For this however, we must first define a notion of uniqueness in the decomposition. Definition 2.15 Two .H-valued processes .Ψ, .Ф are said to be indistinguishable if there exists a set .A ∈ F with .P(A) = 1 such that at all .t ≥ 0 and all .ω ∈ A, .Ψ t (ω) = Фt (ω). As we have become accustomed to, this notion can be built up from the finite dimensional projections and one dimensional theory. Lemma 2.3 Let .Ψ and .Ф be .H-valued processes. Then .Ψ is indistinguishable from .Ф if and only if for every basis vector .ei , .〈Ψ, ei 〉H is indistinguishable from .〈Ф, ei 〉H . Proof The first implication is trivial so we consider only the reverse one. That is, assume that for every .ei there exists a set .Ai ∈ F with .P(Ai ) = 1, and for all .t ≥ 0 and .ω ∈ Ai , 〈Ψ t (ω), ei 〉H = 〈Фt (ω), ei 〉H .

.

We now define .A := .ω ∈ A, .t ≥ 0,

⋂ i

Ai , which is again of full probability and in .F, and for any

‖Ψ t (ω) − Фt (ω)‖2H =

∞ ⎲

.

〈Ψ t (ω) − Фt (ω), ei 〉2H = 0,

i=1

⨆ ⨅

which completes the proof.

Revisiting the quadratic variation, our question is: Can we show that .‖M‖2H defines a submartingale? We have integrability by definition, and 2 .‖Mt ‖ H

=

∞ ⎲

〈Mt , ei 〉2H ,

i=1

where the limit is defined .P − a.s.. Again by definition the above projections are martingales and so the squares are submartingales. The process .‖M‖2H is adapted as each .‖Mt ‖2H is the .P − a.s. limit of .Ft -measurable random variables (on the complete measure space), and it is a submartingale as we can apply the Monotone Convergence Theorem to take the limit through the expectation for the defining submartingale property. As such, we have the following: Definition 2.16 For .M ∈ M2c (H), the quadratic variation .[M] of M is defined to be the unique4 , continuous, adapted, nondecreasing process with .[M]0 = 0 (.P − a.s.)

4 Uniqueness

here is “up to indistinguishability,” as defined in Definition 2.15.

32

2 Stochastic Calculus in Infinite Dimensions

specified in the Doob–Meyer decomposition (Theorem A.1) such that ‖M‖2H − [M]

.

is a real valued martingale. Proposition 2.10 Suppose that .Ψ ∈ IH , then ⎡⎰

⎤

·

⎰

0

t

=

Ψ r dWr

.

t

0

‖Ψ r ‖2H dr.

(2.24)

Proof The fact that the process in (2.24) is continuous, adapted, nondecreasing, and starting from zero is clear. It simply remains to show the required martingality. To this end observe that at each time t, ‖⎰ t ‖2 ⎰ t ‖ ‖ ‖ ‖ − ‖Ψ r ‖2H dr . Ψ dW r r ‖ ‖ H

0

0

∞ /⎰ t ⎲

=

i=1

i=1 ∞ ⎲

=

i=1

Ψ r dWr , ei

0

∞ ⎛⎰ ⎲

=

\2 H

−

⎰ t⎲ ∞ 0 i=1

⎞2

t

〈Ψ r , ei 〉H dWr

0

⎛⎛⎰ 0

−

〈Ψ r , ei 〉2H dr

∞ ⎰ ⎲ 0

i=1 t

⎞2 〈Ψ r , ei 〉H dWr

⎰ − 0

t

t

〈Ψ r , ei 〉2H dr ⎞

〈Ψ r , ei 〉2H dr

having applied Theorem 2.1 to the first term and the Monotone Convergence Theorem to the second term, where the infinite sum is a limit taken .P − a.s.. From the standard one dimensional theory, for each n, ⎛⎛⎰ n ⎲ .

i=1

0

t

⎞2 〈Ψ r , ei 〉H dWr

⎰ − 0

t

⎞ 〈Ψ r , ei 〉2H dr

is a real valued martingale so to conclude the proof we only need to justify that the .P − a.e. limit also holds in .L1 (Ω; R) as convergence in this space preserves martingality. This is a straightforward application of the Monotone Convergence Theorem applied to each integral separately, which concludes the proof. ⨆ ⨅ We also look to reconcile this definition with the one often stated in the one dimensional case, as a limit in probability over any time partition with mesh approaching zero. Proposition 2.11 Let .Ψ ∈ IH T and consider any sequence of partitions

2.4 Martingale Theory in Hilbert Spaces

33

{ } Il := 0 = t0l < t1l < · · · < tkl l = T

.

with .maxj |tjl − tjl −1 | → 0 as .l → ∞. Then for all .t ∈ [0, T ], for any .ε > 0, ⎫⎞ | ⎛⎧| | ‖ ‖2 | ⎪ ⎪ ⎰ t ⎬ | ‖ ⎨| ⎲ ‖⎰ tjl +1 ⎟ | ‖ ‖ ⎜ | ‖Ψ r ‖2H dr | > ε ⎠ = 0. . lim P ⎝ | Ψ r dWr ‖ − ‖ l ⎪ | ‖ ‖ | ⎪ l→∞ 0 ⎭ ⎩|t l ≤t tj | H j +1 We prove this result with an intermediary lemma, stated in the setting of Proposition 2.11. Lemma 2.4 Define the sequence of stopping times .(τ n ) at every .n ∈ N by ⎧

⎫ ‖2 ‖⎰ t ⎰ t ‖ ‖ n ‖ ‖Ψ r ‖2H dr ≥ n .τ := n ∧ inf t ∈ [0, T ] : ‖ ‖ Ψ r dWr ‖ + 0

H

0

and the process .Ψ n· := Ψ · 1·≤τ n . Suppose that for every n and all .t ∈ [0, T ], |⎞ ⎛| | ‖ ‖2 | ⎰ t | ‖ | ⎲ ‖⎰ tjl +1 ‖ ‖ 2 ‖ ‖ ⎜| n n ‖Ψ ‖ dr ||⎟ . lim E ⎝| Ψ r dWr ‖ − ‖ ⎠ = 0. r H | ‖ tl ‖ |l l→∞ 0 j | |tj +1 ≤t H

(2.25)

Then for any .ε > 0, ⎫⎞ | ⎛⎧| | ‖ ‖2 | ⎪ ⎪ ⎰ t ⎬ | ‖ ⎨| ⎲ ‖⎰ tjl +1 ⎟ | ‖ ‖ ⎜ | 2 ‖Ψ r ‖H dr | > ε ⎠ = 0. . lim P ⎝ | Ψ r dWr ‖ − ‖ l ⎪ | ‖ ‖ | ⎪ l→∞ 0 ⎭ ⎩|t l ≤t tj | H j +1 Proof Define ⎫ ⎧| | | | ‖ ‖2 ⎪ ⎪ ⎰ t ⎬ | ‖ ⎨| ⎲ ‖⎰ tjl +1 | | ‖ ‖ l 2 ‖Ψ r ‖H dr | > ε .A := Ψ r dWr ‖ − | ‖ ⎪ | ‖ ⎪ 0 ⎭ ⎩||t l ≤t ‖ tjl | H j +1 and then for any n, Al = Al ∩

.

In particular,

⎡{

} { }⎤ ⎡ { }⎤ ⎡ { }⎤ τ n > T ∪ τ n ≤ T = Al ∩ τ n > T ∪ Al ∩ τ n ≤ T ⎡ { }⎤ { } ⊂ Al ∩ τ n > T ∪ τ n ≤ T .

34

2 Stochastic Calculus in Infinite Dimensions

⎛ ⎛ ⎞ { }⎞ ({ }) P Al ≤ P Al ∩ τ n > T + P τ n ≤ T ,

.

(2.26)

where ⎫ ⎧| | | | ‖⎰ l ‖2 ⎪ ⎪ ⎰ t ⎬ | | ‖ ‖ ⎨ t ⎲ ‖ ‖ { n } | | ‖ j +1 n ‖ l n ‖2 ‖ Ψ r H dr | > ε . := A ∩ τ > T = | Ψ r dWr ‖ − ‖ ⎪ | ‖ ⎪ 0 ⎭ ⎩||t l ≤t ‖ tjl | H j +1

Al,n

.

We then take the limit as .l → ∞ in (2.26), which holds for all n so it must hold in the limit as .n → ∞, providing that .

⎛ ⎞ ⎛ { }⎞ ({ }) lim P Al ≤ lim lim P Al ∩ τ n > T + lim P τ n ≤ T

l→∞

n→∞ l→∞

⎛

{ = lim lim P Al ∩ τ n > T n→∞ l→∞

}⎞

n→∞

(2.27)

given that the .(τ n ) approach infinity .P − a.s.. From Chebyshev’s inequality and the assumption (2.25), .

⎛ { }⎞ lim lim P Al ∩ τ n > T

n→∞ l→∞

|⎞ ⎛| | ‖⎰ l ‖2 ⎰ | ‖ ‖ | t t ⎲ ‖ n ‖2 ||⎟ 1 ‖ j +1 n ‖ ⎜| ‖Ψ ‖ dr |⎠ = 0, lim lim E ⎝| ≤ Ψ r dWr ‖ − ‖ r H | ‖ tl ‖ |l ε n→∞ l→∞ 0 j | |tj +1 ≤t H

which combined with (2.27) gives the result.

⨆ ⨅

Proof of Proposition 2.11 We once again look to prove this by considering the finite dimensional projections on which the result is known to be true, before showing that it is preserved in the limit. This approach is ultimately taken for the localized sequence by applying Lemma 2.4, looking to verify (2.25). Identically to the proof of Proposition 2.10, we have that |⎞ ⎛| | ‖ ‖2 | ⎰ t | ‖ | ⎲ ‖⎰ tjl +1 ‖ ‖ 2 ‖ ‖ ⎜| ‖Ψ n ‖ dr ||⎟ .E ⎝| Ψ nr dWr ‖ − ‖ ⎠ r H | ‖ l ‖ |l 0 | |tj +1 ≤t tj H |⎞ ⎛| | | ⎛⎰ l ⎞2 ⎰ ∞ |⎲ ⎲ tj +1 〈 t 〉 〈 n 〉2 ||⎟ ⎜| n = E ⎝| Ψ r , ei H dWr − Ψ r , ei H dr |⎠ l |l | 0 |tj +1 ≤t i=1 tj |

2.4 Martingale Theory in Hilbert Spaces

35

|⎞ | | | ⎛⎰ l ⎞2 ⎰ t t 〉 〈 n 〉2 ||⎟ j +1 〈 |⎲ ⎜ n Ψ r , ei H dWr − Ψ r , ei H dr |⎠ ≤ E⎝ | | | tjl 0 i=1 |t l ≤t | j +1 |⎞ | ⎛ | | ⎛⎰ l ⎞2 ⎰ ∞ | tj +1 〈 t〈 ⎲ ⎜| ⎲ 〉 〉2 ||⎟ n n = Ψ r , ei H dWr − Ψ r , ei H dr |⎠ , E ⎝| l | |l 0 i=1 | |tj +1 ≤t tj ⎛

∞ | ⎲

where on the last line we have applied the Monotone Convergence Theorem. From Theorem A.3 we know that for each .i ∈ N, ⎛| ⎤||⎞ ⎡⎛ | ⎞2 ⎰ ⎰ tl | |⎲ t〈 〈 〉 〉 j +1 2 |⎟ ⎜| ⎣ Ψ nr , ei H dWr − Ψ nr , ei H dr ⎦|⎠ = 0, . lim E ⎝| l | |l l→0 tj 0 |tj +1 ≤t | so we would be done if we can justify the interchange of infinite sum and limit in l. We look to apply the Dominated Convergence Theorem to proceed, noting that for each fixed .i, l ∈ N, ⎛| ⎤||⎞ ⎡⎛ | ⎞2 ⎰ ⎰ tl | |⎲ t 〉 〈 n 〉2 j +1 〈 |⎟ ⎜| ⎣ Ψ nr , ei H dWr − Ψ r , ei H dr ⎦|⎠ .E ⎝| l | |l tj 0 | |tj +1 ≤t ⎤ ⎡⎛ ⎞2 ⎡⎰ t ⎤ ⎰ tl ⎲ 〉 〈 n 〉2 j +1 〈 n ≤ Ψ r , ei H dWr ⎦ + E Ψ r , ei H dr E⎣ tjl

tjl +1 ≤t

=

⎲

⎡⎰ E

tjl +1 ≤t

⎡⎰

= 2E 0

t

tjl +1 tjl

0

⎤ ⎡⎰ t ⎤ 〈 n 〉2 〈 n 〉2 Ψ r , ei H dr + E Ψ r , ei H dr

⎤ 〈 n 〉2 Ψ r , ei H dr ,

0

(2.28)

⎡⎰ 〈 〉2 ⎤ t which is a bound uniform in l and summable in i. Thus, .2E 0 Ψ nr , ei H dr is our dominating function which justifies the application of the Dominated Convergence Theorem, concluding the verification of (2.25) and hence the proof by Lemma 2.4. ⨆ ⨅ Lemma 2.5 Suppose that .(M n ) is( a sequence of martingales in .M2c (H) which at ) 2 every time .t ≥ 0 converges in .L Ω; H to some .Mt . Suppose in addition that at ( ) any time .t ≥ 0, the sequence .([M n ]t ) converges to some .ψt in .L1 Ω; R where .ψ is continuous, adapted, and nondecreasing (.P − a.s.). Then .M ∈ M2c (H) and .[M] is indistinguishable from .ψ.

36

2 Stochastic Calculus in Infinite Dimensions

Proof The fact that .M ∈ M2c (H) is immediate from Proposition 2.7, so we move on to the quadratic variation. Observe that for each n, by definition ‖ n ‖2 ‖M ‖ − [M n ] H

.

( ) is a real valued martingale, so the .L1 Ω; R limit at each time t, .

(‖ ‖2 ) lim ‖M n ‖H − [M n ]

n→∞

is again a real valued martingale if it exists. But this is clear as we have that .

‖ ‖2 lim ‖M n ‖H − lim [M n ] = ‖M‖2H − ψ

n→∞

n→∞

by definition of the limit. From Definition 2.16, then if .ψ0 = 0 .P−a.s., we have that ψ satisfies the conditions of the quadratic variation of M, so it is indistinguishable from it. This property is immediate as .ψ0 is the limit in .L1 (Ω; R) of .([M n ]0 ) by assumption, while this is just a sequence of zeros .P − a.s., hence the result. ⨆ ⨅

.

In part due to this result, we provide an alternative characterization of the quadratic variation under some additional boundedness assumptions. We reintroduce the projections .(Pk ) used in Proposition 2.7. Proposition 2.12 For any .M ∈ M2c (H), .[M] has the representation [M] =

.

∞ ⎲ ⎡ ⎤ 〈M, ei 〉H i=1

P − a.s. for the limit taken in .C ([0, T ]; R) for any .T ≥ 0.

.

Proof Observe that ‖Pk M‖2H =

k ⎲

.

〈M, ei 〉2H ,

i=1

which is the sum of k one dimensional submartingales, and in particular

.

k ⎛ ⎲ ⎤⎞ ⎡ 〈M, ei 〉2H − 〈M, ei 〉H i=1

is a martingale. From the definition of the quadratic variation, it is thus clear that

2.4 Martingale Theory in Hilbert Spaces

.

[Pk M] =

37 k ⎲ ⎡ ⎤ 〈M, ei 〉H .

(2.29)

i=1

Moreover we have that at each .t ≥ 0, .Pk Mt is convergent to .Mt .P − a.s. in .H, the Monotone Convergence and furthermore in .L2 (Ω; H) from an application of ∑ 2 Theorem to the difference process .‖M − Pk M‖2H = ∞ i=k+1 〈M, ei 〉H . Reminiscent of Lemma 2.5, we look to show convergence of the sequence .([Pk M]) to some 1 .ψ at each .t ≥ 0 in .L (Ω; R), and in fact we show the stronger convergence in 1 .L (Ω; C ([0, T ]; R)) for each .T ≥ 0. We proceed by showing the Cauchy property in this Banach Space. To do this we consider, for .j < k, [Pk M] − [Pj M] =

.

k ⎲ ⎡ ⎤ 〈M, ei 〉H .

(2.30)

i=j +1

From this identity and the preceding work, it is clear that [Pk M] − [Pj M] = [Pk M − Pj M]

(2.31)

.

and therefore .

| | sup |[Pk M]t − [Pj M]t | = sup [Pk M −Pj M]t = [Pk M −Pj M]T .

t∈[0,T ]

t∈[0,T ]

(2.32)

Furthermore, we are concerned with a control in expectation of this term. From the property that real valued martingales have constant expectation, we deduce that ⎞ ⎛‖ ‖2 E ‖Pk MT − Pj MT ‖H − [Pk M − Pj M]T ⎞ ⎛‖ ‖2 = E ‖Pk M0 − Pj M0 ‖H − [Pk M − Pj M]0 ⎛‖ ‖2 ⎞ = E ‖Pk M0 − Pj M0 ‖

.

H

so in particular ⎛‖ ‖ ‖2 ‖2 ⎞ ) ( E [Pk M − Pj M]T = E ‖Pk MT − Pj MT ‖H − ‖Pk M0 − Pj M0 ‖H ⎛ ⎞ k ⎛ ⎞ ⎲ 〈MT , ei 〉2H − 〈M0 , ei 〉2H ⎠ = E⎝

.

⎛ ≤ E⎝

i=j +1 k ⎲

i=j +1

⎞

⎛

〈MT , ei 〉2H ⎠ + E ⎝

k ⎲ i=j +1

⎞ 〈M0 , ei 〉2H ⎠ .

38

2 Stochastic Calculus in Infinite Dimensions

By(∑ the square integrability assumption on M, we have that at every .t ≥ 0, ) ∞ 2 〈M 〉 < ∞, E , e which justifies that the right hand side of the above t i i=1 H approaches zero as .j → ∞, uniformly in k. With (2.32) we deduce that the sequence 1 .([Pk M]) is Cauchy in .L (Ω; C ([0, T ]; R)) for each .T ≥ 0 so it admits a limit in this space which we call .ψ. Through the .L1 (Ω; C ([0, T ]; R)) convergence, we can deduce the existence of a subsequence which is .P−a.s. convergent in .C ([0, T ]; R). In fact we can upgrade this to convergence over the whole sequence, and as was noted that .P − a.s. at each .t ≥ 0 the real valued sequence .([Pk M]t ) not only has a convergent subsequence but is nondecreasing in k, which implies convergence of the whole sequence. Convergence of the whole sequence .P − a.s. in .C ([0, T ]; R) can then be deduced by the Cauchy property with (2.32) and (2.31). In summary thus far we have that .

ψ=

.

∞ ⎲ ⎡ ⎤ 〈M, ei 〉H

P − a.s.

i=1

for the limit (of the sequence of finite sums) taken in .C ([0, T ]; R) for any .T ≥ 0. It only remains to show that .ψ is indistinguishable from .[M], which we do by verifying the conditions of Lemma 2.5. The convergence has already been established; hence we need only the regularity on .ψ. It is continuous by construction and must be adapted as .ψt is the .P − a.s. limit of the .Ft -measurable .[Pk M]t on the complete measure space. This limit similarly preserves the nondecreasing property, so .ψ satisfies the required conditions and must be indistinguishable from .[M] due ⨆ ⨅ to Lemma 2.5. Following on from the quadratic variation, we look to introduce the crossvariation between martingales. Guided by the motivation to use the Stratonovich integral, introduced in Sect. 3.1, we will need to consider the cross-variation between elements of .M2c (H) and .M2c . Defining this akin to a polarization identity is out of the question as one cannot take sums of these martingales (there is no canonical way to sum an element of a Hilbert space with an element of .R), so we look to use the characterization in terms of their product. Indeed from the classical theory, Theorem A.2, for any given .Ψ ∈ M2c (H), .Y ∈ 2 Mc , and .ei a basis vector of .H, there exists a unique continuous, adapted, boundedvariation process .[〈Ψ, ei 〉H , Y ] with .[〈Ψ, ei 〉H , Y ]0 = 0 (.P − a.s.) such that 〈Ψ, ei 〉H Y − [〈Ψ, ei 〉H , Y ]

.

is a real valued martingale. We would love to immediately have the existence and uniqueness of a corresponding .H-valued process which gives a martingale when subtracted from .ΨY , but that is not clear in the same way that the quadratic variation was as in that case .‖Ψ‖2H was a genuine real valued submartingale. Our approach, therefore, comes from the characterization in Proposition 2.12. We make a first definition for the projected process.

2.4 Martingale Theory in Hilbert Spaces

39

Definition 2.17 For .Ψ ∈ M2c (H) and .Y ∈ M2c , for any .k ∈ N, we define the cross-variation process .[Pk Ψ, Y ] by [Pk Ψ, Y ] :=

k ⎲

.

[〈Ψ, ei 〉H , Y ]ei .

i=1

Indeed we can verify that such a process gives us the desired properties of a cross-variation in .H. Proposition 2.13 For .Ψ ∈ M2c (H) and .Y ∈ M2c , for any .k ∈ N, .[Pk Ψ, Y ] is the unique continuous, adapted, bounded-variation .H-valued process satisfying .[Pk Ψ, Y ]0 = 0 .P − a.s. such that (Pk Ψ)Y − [Pk Ψ, Y ]

.

is an .H-valued martingale. Proof It follows from the corresponding properties of the real valued crossvariations that .[Pk Ψ, Y ] is continuous, adapted, and of bounded-variation (one can apply the triangle inequality for the norm .‖·‖H ) satisfying .[Pk Ψ, Y ]0 = 0 P − a.s.. In addition observe that (Pk Ψ)Y − [Pk Ψ, Y ] =

.

k ⎲ ( ) 〈Ψ, ei 〉H Y − [〈Ψ, ei 〉H , Y ] ei , i=1

which we look to show it is an .H-valued martingale. To this end consider arbitrary φ ∈ H. Then

.

〈(Pk Ψ)Y − [Pk Ψ, Y ], φ〉H / k \ ∞ ⎲( 〈 〉 ) ⎲ 〈Ψ, ei 〉H Y − [〈Ψ, ei 〉H , Y ] ei , φ, ej H ej =

.

i=1

=

∞ k ⎲ ⎲ i=1 j =1

=

j =1

H

〈( 〉 〉 ) 〈 〈Ψ, ei 〉H Y − [〈Ψ, ei 〉H , Y ] ei , φ, ej H ej H

k ⎲ ( ) 〈Ψ, ei 〉H Y − [〈Ψ, ei 〉H , Y ] 〈φ, ei 〉H , i=1

where we recall that each process .〈Ψ, ei 〉H Y − [〈Ψ, ei 〉H , Y ] is a martingale, hence too(is ) . 〈Ψ, ei 〉H Y − [〈Ψ, ei 〉H , Y ] 〈φ, ei 〉H , and therefore the finite sum is a martingale.

40

2 Stochastic Calculus in Infinite Dimensions

To comment on the uniqueness of .[Pk Ψ, Y ], we invoke Lemma 2.3. Suppose that Π is an .H-valued process which is continuous, adapted, and of bounded-variation satisfying .Π0 = 0 .P − a.s.. Moreover, suppose that

.

(Pk Ψ)Y − Π

.

is an .H-valued martingale. Take any basis vector .ej . Then 〈 〉 (Pk Ψ)Y − Π, ej H

.

is a real valued martingale, but this is just k ⎲ .

〈 〉 〉 〈 〈Ψ, ei 〉H ei , ej H Y − Π, ej H .

(2.33)

i=1

〈 〉 From the regularity of .Π, we have again that . Π, ej H is continuous, adapted, 〈 〉 and of bounded-variation satisfying . Π0 , ej H = 0 .P − a.s.. There are two cases here: .j ≤ k and .j > k.〈 In the 〈 the 〉 first term 〉 latter case we have that in (2.33) is null so in particular . Π, ej H is a martingale. Thus . Π, ej H ∈ Mc and is of bounded-variation, so by Lemma A.1 it is .P − a.s. constant. As the 〈process〉 starts from zero, then it is indistinguishable 〈 〉 from the null process. Therefore . Π, ej is indistinguishable from . [Pk Ψ, Y ], ej , which is similarly zero (recall H Definition 2.17). In the alternative case .j ≤ k, the real valued martingale (2.33) is given by 〈 〉 〉 〈 Ψ, ej H Y − Π, ej H .

.

〈 〉 Therefore . Π, ej H satisfies the requirements of the cross-variation process 〉 〈 .[ Ψ, ej , Y ] and hence is indistinguishable from it. This, however, is simply H .〈[Pk Ψ, Y ], ek 〉 from its definition. Combining with Lemma 2.3, the theorem is ⨆ ⨅ proven. It remains for us to define the cross-variation process .[Ψ, Y ]. Referencing Lemma A.2, we know that .P − a.s. for any .t ≥ 0, [〈Ψ, ei 〉H , Y ]2t ≤ [〈Ψ, ei 〉H ]t [Y ]t ,

.

hence, using the nondecreasing property of quadratic variation, .

sup [〈Ψ, ei 〉H , Y ]2t ≤ sup

t∈[0,T ]

t∈[0,T ]

( ) [〈Ψ, ei 〉H ]t [Y ]t

⎛

≤

sup [〈Ψ, ei 〉H ]t

t∈[0,T ]

⎞⎛

⎞ sup [Y ]t

t∈[0,T ]

2.4 Martingale Theory in Hilbert Spaces

41

= [〈Ψ, ei 〉H ]T [Y ]T . Moreover for .j < k, ‖ ‖2 ‖ k ‖ ‖ ⎲ ⎡ ‖ ‖2 ⎤ ‖ ‖ ‖ ‖ 〈Ψ, ei 〉H , Y t ei ‖ [Pk Ψ, Y ]t − [Pj Ψ, Y ]t H = sup ‖ . sup ‖ t∈[0,T ] t∈[0,T ] ‖i=j +1 ‖

H

= sup

k ⎲

t∈[0,T ] i=j +1

≤

≤

k ⎲

sup

i=j +1 t∈[0,T ] k ⎲

⎤2 ⎡ 〈Ψ, ei 〉H , Y t ⎤2 ⎡ 〈Ψ, ei 〉H , Y t

[〈Ψ, ei 〉H ]T [Y ]T

i=j +1

= [Pk Ψ − Pj Ψ]T [Y ]T using (2.30) and (2.31) for the last line. We look to follow a similar approach to Proposition 2.12, taking the expectation and showing a Cauchy property. It is unknown if this resulting term is integrable, so in order to take the expectation we introduce the localizing times τn := n ∧ inf {0 ≤ t < ∞ : [Y ]t ≥ n} .

.

Then ⎛⎛ E⎝

.

‖ ‖ sup ‖[Pk Ψ, Y ]t 1t≤τn − [Pj Ψ, Y ]t 1t≤τn ‖H

⎞2 ⎞ ⎠

t∈[0,T ]

⎛ =E (

‖ ‖2 sup ‖[Pk Ψ, Y ]t 1t≤τn − [Pj Ψ, Y ]t 1t≤τn ‖H

t∈[0,T ]

≤ E [Pk Ψ − Pj Ψ]T [Y ]T 1t≤τn ( ) ≤ nE [Pk Ψ − Pj Ψ]T ,

⎞

)

which was shown to approach zero as .j → ∞, uniformly in k, in Proposition 2.12. that for every .n ∈ N and .T ≥ 0, the sequence (We have thus demonstrated ) 2 ∞ ([0, T ]; H)). We can . [Pk Ψ, Y ]· 1·≤τn is Cauchy in the Banach Space .L (Ω; L therefore extract a subsequence which converges .P − a.s. in .L∞ ([0, T ]; H). To remove the truncation we introduce the sets

42

2 Stochastic Calculus in Infinite Dimensions

An := {ω ∈ Ω : τn (ω) ≥ T } .

.

Then on every .An , there exists a subsequence of .([Pk Ψ, Y ]) which is .P − a.s. (within .An ) convergent in .L∞ ([0, T ]; H). It should be noted that the choice of subsequence may be dependent on n, hence the separation, and also that this convergence is now of continuous processes so can be taken in .C ([0, T ]; H). We look to upgrade this convergence to be of the whole sequence, by another Cauchy argument. Fix any .ε > 0, and we look to show the existence of a .J ∈ N such that for all .k ≥ J , sup ‖[Pk Ψ, Y ]t − [PJ Ψ, Y ]t ‖2H < ε

.

t∈[0,T ]

or, equivalently, as already shown in the proof,

.

k ⎲ ⎤2 ⎡ 〈Ψ, ei 〉H , Y t < ε.

sup

t∈[0,T ] i=J +1

Let the convergent subsequence be indexed by .km . This convergence implies that the subsequence is Cauchy, so there exists a .Jm such that for all .km > Jm , .

‖ ‖2 sup ‖[Pkm Ψ, Y ]t − [PJm Ψ, Y ]t ‖H < ε

t∈[0,T ]

or equivalently

.

km ⎲ ⎡ ⎤2 〈Ψ, ei 〉H , Y t < ε.

sup

t∈[0,T ] i=J +1 m

We now set .J := Jm and argue that for every .k > J there exists a .km > k such that

.

sup

k ⎲ ⎡ ⎤2 〈Ψ, ei 〉H , Y t ≤ sup

t∈[0,T ] i=J +1

km ⎲ ⎡ ⎤2 〈Ψ, ei 〉H , Y t < ε

t∈[0,T ] i=J +1

which proves the Cauchy property. Hence on each .An , the sequence .([Pk Ψ, Y ]) is P − a.s. convergent in .C ([0, T ]; H). We use that

.

⎛ P

⋃

.

n

⎞ An

⎛ =1−P

⋂

⎞ Acn

⎛ =1−P

⋂

n

⎞ {ω ∈ Ω : τn (ω) < T }

n

= 1 − lim P ({ω ∈ Ω : τn (ω) < T }) n→∞

2.4 Martingale Theory in Hilbert Spaces

43

owing to the fact that the .(τn ) are .P − a.s. increasing. Moreover, this limit is zero given that the .(τn ) approach infinity. Thus we obtain the .P − a.s. convergence on .Ω. Definition 2.18 For .Ψ ∈ M2c (H) and .Y ∈ M2c , we define the cross-variation process .[Ψ, Y ] by [Ψ, Y ] :=

∞ ⎲

.

[〈Ψ, ei 〉H , Y ]ei

i=1

P − a.s. for the limit taken in .C ([0, T ]; H).

.

Our next question is then very natural: Do we have the corresponding characterization as in Proposition 2.13? The fact that this cross-variation is continuous, adapted, and starting from zero is proven as in Proposition 2.12. In fact the martingality of ΨY − [Ψ, Y ]

(2.34)

.

is again proven near identically, as we use that (Pk Ψ)Y −

.

k ⎲ [〈Ψ, ei 〉H , Y ]ei i=1

is a martingale, and the convergence of this at each .t ≥ 0 in .L1 (Ω; H) to (2.34). It is the bounded-variation which proves problematic. Possibly the most logical approach is to show that the finite sums are of uniformly bounded total variation, through an argument like T .V H

([Pk Ψ, Y ]) =

VHT

⎛ k ⎲

⎞ [〈Ψ, ei 〉H , Y ]ei

≤

i=1

k ⎲

( ) VHT [〈Ψ, ei 〉H , Y ]ei

i=1

=

k ⎲

) ( VRT [〈Ψ, ei 〉H , Y ] .

i=1

Then from Lemma A.3, we have that ( ) ) 1( [〈Ψ, ei 〉H ]T + [Y ]T , VRT [〈Ψ, ei 〉H , Y ] ≤ 2

.

∑ but this means taking . ki=1 [Y ]T , which will explode as .k → ∞. In lieu of this bounded-variation, we do of course have the weaker property that every projection is of bounded-variation. In fact, this is enough for uniqueness.

44

2 Stochastic Calculus in Infinite Dimensions

Suppose that .Π is an .H-valued process which is〈continuous, adapted, satisfying 〉 Π0 = 0 and such that for every basis vector .ej , . Π, ej is of bounded-variation .P − a.s.. Moreover suppose that .

ΨY − Π

.

is an .H-valued martingale. Take any basis vector .ej . Then 〈 〉 ΨY − Π, ej H

.

is a real valued martingale, but this is just ∞ ⎲ .

〈 〉 〉 〈 〈Ψ, ei 〉H ei , ej H Y − Π, ej H

i=1

or simply 〈 〈 〉 〉 Ψ, ej H Y − Π, ej H .

.

〈 〉 〉 〈 Thus . Π, ej H is indistinguishable from .[ Ψ, ej H , Y ] which is simply .〈[Ψ, Y ], ek 〉 from its definition. Combining with Lemma 2.3, we deduce the uniqueness in this class, so have proven the following: Proposition 2.14 For .Ψ ∈ M2c (H) and .Y ∈ M2c , .[Ψ, Y ] is the unique, continuous, adapted .H-valued process satisfying .[Ψ, Y ]0 = 0 .P − a.s. such that for every basis 〈 〉 vector .ej , . [Ψ, Y ], ej H is of bounded-variation .P − a.s. and ΨY − [Ψ, Y ]

.

is an .H-valued martingale. We now state and prove the analogous result to Lemma 2.5, which will be necessary in the Itô–Stratonovich conversion. Lemma 2.6 Suppose that .(Ψ n ) is a sequence of martingales in .M2c (H) which at ( ) every time .t ≥ 0 converges in .L2 Ω; H to some .Ψ t . Let .Y ∈ M2c . Suppose in n addition ( )that at any time .t ≥ 0, the sequence .([Ψ , Y ]t ) converges to some .Lt in 1 .L Ω; R , where L is a continuous, adapted process, and for every basis vector 〈 〉 .ej , . L, ej is of bounded-variation .P − a.s.. Then .Ψ ∈ M2c (H) and .[Ψ, Y ] is H indistinguishable from L. Proof For each n the martingale in question is the .H-valued one Ψ n Y − [Ψ n , Y ].

.

2.5 Integration with Respect to Cylindrical Brownian Motion

45

( ) We use again that the .L1 Ω; H limit preserves martingality and that by Hölder’s ( ) inequality the .L1 Ω; H limit of .Ψ nt Yt is .Ψ t Yt . With the same arguments as in Lemma 2.5 we conclude the proof. ⨆ ⨅ If the given processes were only (continuous) local martingales, then we can make a slightly modified version of the definition. Assuming without loss of generality that .Ψ and Y are locally square integrable (see the discussion after Definition 2.9), localized by stopping times .(Rn ) and .(Tn ), respectively, then for a new sequence of stopping times defined by .τn = Rn ∧Tn , the stopped processes .Ψ τn and .Y τn are genuine square integrable martingales (in their respective spaces), so the cross-variation .[Ψ τn , Y τn ] can be defined. The canonical localization procedure is evident once more, as the .(τn ) tends to infinity almost surely, the consistency conditions that for .m ≤ n and .t ≤ τm we have Ψ τt m = Ψ τt n

.

and

Ytτm = Ytτn

allow us once more to define the process at any t by [Ψ, Y ]t := lim [Ψ τn , Y τn ]t

.

n→∞

(2.35)

for the limit taken .P − a.s. in .H. Then the process ΨY − [Ψ, Y ]

.

is itself a local martingale, localized by the stopping times .(τn ). The argument justifying this is identical to Proposition 2.9, from which it is similarly clear that [Ψ, Y ]τn = [Ψ τn , Y τn ].

.

In the traditional way, these notions can all be extended to semimartingales (that is, a martingale plus a bounded-variation process). The quadratic and crossvariations of such semi-martingales are then simply the quadratic/cross-variation of ¯ 2 and the corresponding martingale parts. To this end we introduce the notations .M c ¯ 2 (H) to be the corresponding spaces of square integrable continuous semimartin.M c ¯ c, M ¯ c (H) to be the spaces of continuous semimartingales. gales, and similarly .M

2.5 Integration with Respect to Cylindrical Brownian Motion For our analysis now we will need to make reference to two distinct Hilbert spaces: one over which .W is a cylindrical Brownian motion, and the other in which our integrand maps to. Henceforth we introduce .U as the Hilbert space over which .W

46

2 Stochastic Calculus in Infinite Dimensions

is a cylindrical Brownian motion. We shall take .(ei ) as an orthonormal basis over .U and .(ai ) an orthonormal basis over .H. Definition 2.19 Denote by .IH operator T (W) the class ( of progressively measurable ) valued processes B belonging to the set .L2 Ω × [0, T ]; L 2 (U; H) . Measurability here is again defined with respect to the Borel sigma algebra on .L 2 (U; H). The H class of processes B such that .B ∈ IH T (W) for all T will be denoted by .I (W). Note that we make no explicit reference to .U, the space on which .W is a cylindrical Brownian motion. This is because, in practice, the space .U will be arbitrarily chosen; this shall be discussed later. Recall (2.22) that if .W is a cylindrical Brownian motion over .U, it can be formally represented by W(t) =

∞ ⎲

.

ei Wti ,

(2.36)

i=1

where the .(W i ) are standard independent one dimensional Brownian motions. Definition 2.20 For .B ∈ IH (W), we define the Itô stochastic integral ⎰

t

(2.37)

B(s)dWs

.

0

as the .H-valued random variable ∞ ⎰ ⎲ .

i=1

0

t

Bei (s)dWsi ,

(2.38)

where ( each ) integral is defined as in Definition 2.6 and the infinite sum is taken in L2 Ω; H .

.

The immediate response to this definition is to prove that (2.38) is well defined; that is,( the integrals are ) well defined, as is the limit. First for ( each i, .Bei is trivially) in .L2 Ω × [0, T ]; H as this norm is bounded by the .L2 Ω × [0, T ]; L 2 (U; H) norm of B. The progressive measurability is inherited from that of B. In order to show that the limit of partial sums is well defined, we proceed similarly to the method applied for (2.19) and argue that the sequence of partial sums is Cauchy. Observe that n ⎰ t || ⎲ ||2 || || .|| Bei (s)dWsi || 2 ( i=m 0

L Ω;H

n ⎰ t || ⎲ ||2 || ) = E|||| Bei (s)dWsi || i=m 0

=

n ⎲ i=m

⎰

t

E 0

‖ ‖ ‖Be (s)‖2 ds i H

H

2.5 Integration with Respect to Cylindrical Brownian Motion

47

having applied 2.3 to the above. But by the assumption ( the Itô isometry Proposition ) that .B ∈ L2 Ω × [0, t]; L 2 (U; H) , we know E

.

⎰ t⎲ ∞ ‖ ‖ ‖Be (s)‖2 ds < ∞ i H 0 i=1

and thus, by Tonelli’s theorem regarding the infinite sum as an integral with respect to the counting measure, ∞ ⎲ .

i=1

⎰

t

E 0

‖ ‖ ‖Be (s)‖2 ds < ∞ i H

demonstrating that the sequence of partial sums is indeed Cauchy in .L2 (Ω; H). Of course the .L2 (Ω; H) norm of the limit is the limit of the .L2 (Ω; H) norms, so we have justified the following. Proposition 2.15 For .B ∈ IH (W), we have ⎛‖⎰ ‖2 ⎞ ⎞ ⎛⎰ t ‖ t ‖ 2 ‖ ‖ ‖B(s)‖ .E B(s)dW ds . = E s 2 ‖ ‖ L (U;H) H

0

0

It is worth noting that while we impose the condition ⎛⎰ E

∞ t⎲ ‖

.

0 i=1

‖ ‖Be (s)‖2 ds i H

⎞ < ∞,

one may instead require the weaker condition ⎰ t⎲ ∞ ‖ ‖ ‖Be (s)‖2 ds < ∞ . i H

P − a.s.

(2.39)

0 i=1

( ) or equivalently that .B : Ω → L2 [0, t]; L 2 (U; H) for .P−a.e. .ω. Our formulation follows the classical construction as laid out in Sect. 2.1, ensuring that the integral is a genuine square integrable martingale, which one anticipates would be lost with just the assumption (2.39). We can just as straightforwardly follow the arguments from Definition 2.9, which are laid out here. The extension is pertinent in applications, and this brief will closely monitor the necessity of localization in the abstract frameworks of Chaps. 3 and 4. Definition 2.21 Denote by .IH T (W) the class of progressively ( measurable operator ) valued processes B such that .B(ω) belongs to the set .L2 [0, T ]; L 2 (U; H) for H .P − a.e. .ω. The class of processes B such that .B ∈ I (W) for all T will be T denoted by .IH (W).

48

2 Stochastic Calculus in Infinite Dimensions

Using the template for Definition 2.10, for a process .B ∈ IH (W), let us introduce ⎰ t ‖B(s)‖2L 2 (U;H) ds ≥ n} .τn := n ∧ inf{0 ≤ t < ∞ : 0

taking the convention that the infimum of the empty set is infinite. The .(τn ) are stopping times as⎰they are simply first hitting times of the continuous and adapted t random variable . 0 ‖B(s)‖2L 2 (U;H) ds. These times tend to infinity .P − a.s. by condition (2.39). Now define the truncated processes .B n as B n (t) := B(t)1t≤τn ,

.

and using the fact that for .m ≤ n, and .t ≤ τm , we have B(t)1t≤τn = B(t)1t≤τm

.

we can make the following consistent definition. Definition 2.22 In the setting described, we define ⎰ .

t

⎰ B(s)dWs := lim

n→∞ 0

0

t

B n (s)dWs

(2.40)

P − a.s. .ω in .H.

.

We will have no reservations in writing (2.40) as a formal expression ∞ ⎰ ⎲ .

i=1

0

t

Bei (s)dWsi .

(2.41)

We( say the ) expression is only formal, as the infinite sum in (2.41) is not the L2 Ω; H limit of the partial sums of the local martingales as presented. We understand( (2.41)) only by (2.40), that is, by the limit as .n → ∞ of the infinite sum in .L2 Ω; H of the genuine square integrable martingales given by stopping the local martingales at .τn . We now look to show a series of properties of this integral which were shown for a one dimensional Brownian motion across the earlier sections. The first is the corresponding result of Theorem 2.2.

.

Proposition 2.16 Suppose that .H1 , H2 are Hilbert spaces such that .B ∈ IH1 (W) and .T ∈ L (H1 ; H2 ). Then the process T B defined by ( ) T Bei (s, ω) = T Bei (s, ω)

.

belongs to .IH2 (W). In addition, we have that

2.5 Integration with Respect to Cylindrical Brownian Motion

⎛⎰

t

T

.

⎰

⎞

B(s)dWs =

49

t

T B(s)dWs .

0

0

Moreover, the two integrals are defined as limits .P − a.s. with respect to the same stopping times, and if .B ∈ IH1 (W), then .T B ∈ IH2 (W). Proof Assume at first that .B ∈ IH1 (W). We shall prove first that .T B ∈ IH2 (W). The progressive measurability is preserved under the continuity of T , and letting C be such that for any .φ ∈ H1 , ‖T φ‖2H2 ≤ C‖φ‖2H1 ,

.

which exists owing to the boundedness of T , we have ⎰ t⎲ ⎰ t⎲ ∞ ∞ ‖2 ‖ ‖ ‖ ‖ ‖ ‖Be (s)‖2 ds < ∞ T Bei (s) H ds ≤ C . i H 2

0 i=1

(2.42)

1

0 i=1

holding .P − a.s. as .B ∈ IH1 (W). In addition for any stopping time .τn as in Definition 2.22, E

.

∞ ∞ ⎛⎰ t ⎲ ⎞ ⎛⎰ t ⎲ ⎞ ‖( ‖ ‖ ( )‖ ) ‖ T Be (s) 1s≤τ ‖2 ds = E ‖T Be (s)1s≤τ ‖2 ds n H n i i H 2

0 i=1

2

0 i=1

∞ ⎞ ⎛⎰ t ⎲ ‖ ‖ ‖Be (s)1s≤τ ‖2 ds ≤ CE n i H 0 i=1

1

< ∞. So the new stochastic integral ⎰

t

T B(s)dWs

.

0

can be constructed using the same sequence of stopping times. We will freely use the linearity of T to commute it with the indicator function. To carry T through the integral, it is sufficient to show that for .B n as in Definition 2.22, then T

∞ ⎰ ⎛⎲

t

.

i=1

0

Beni (s)dWsi

⎞

=

∞ ⎰ ⎲ i=1

0

t

T Beni (s)dWsi ,

(2.43)

( ) ( ) where the left hand side limit is taken in .L2 Ω; H1 and the right side in .L2 Ω; H2 . ( ) From the .L2 Ω; H1 limit there exists a subsequence convergent .P − a.e. in .H1 . Working with this subsequence, we can pass the .P − a.s. limit through the

50

2 Stochastic Calculus in Infinite Dimensions

continuous T such that it is now the .P − a.s. limit in .H2 . Linearity of T allows us to pass through the summation as well, so we have now that T

∞ ⎰ ⎛⎲

.

i=1

0

t

nk ⎞ ⎞ ⎛⎰ t ⎲ Beni (s)dWsi = lim T Beni (s)dWsi nk →∞

(2.44)

0

i=1

for the limit .P − a.e. of the subsequence indexed by .(nk ). Applying Theorem 2.2, we can commute T with the integral on the right side of (2.44). we ( However ) have justified already that the limit over the whole sequence in .L2 Ω; H2 exists, ( ) agreeing with the .L2 Ω; H2 limit of the subsequence, which in turn agrees with the .P − a.s. limit. Thus (2.43) is verified, completing the proof. In the case where H H .B ∈ I 1 (W), it is clear from (2.42) that .T B ∈ I 2 (W). ⨆ ⨅ H Proposition 2.17 Let .B ∈ I¯ (W) and .φ : Ω → H be .F0 -measurable. Then for every .t > 0 we have that

/⎰

\

t

Br dWr , φ

.

0

⎰ H

t

= 0

〈Br , φ〉H dWr

(2.45)

H P − a.s.. Moreover if .η : Ω → R is .F0 -measurable, then .ηB ∈ I¯ (W), and for every .t > 0 we have that

.

⎰ η

.

0

t

⎰ Br dWr =

t

ηBr dWr

(2.46)

0

P − a.s..

.

Proof First we make clear that .〈B, φ〉H is understood as a process defined by the mapping (ei , ω, t) → 〈Bt (ei , ω), φ(ω)〉H .

.

R The fact that .〈B, φ〉H ∈ I¯ (W) is completely analogous to Proposition 2.4, where the progressive measurability follows as the mapping

〈B· , φ〉H : (t, ω, ω) ˜ → 〈Bt (ω), φ(ω)〉 ˜ H

.

is .B([0, T ]) × FT × F0 measurable as a mapping into .L 2 (U; R). Similarly we have that ‖ ‖ ‖〈Bt (ω), φ(ω)〉H ‖ 2 ‖φ(ω)‖H ‖Bt (ω)‖L 2 (U;H) , L (U;R) ≤

.

R which is sufficient to justify that .〈B, φ〉H ∈ I¯ (W). The extension of Proposition 2.5 to this result is then identical to the extension of Theorem 2.2 to 2.16, so we

2.5 Integration with Respect to Cylindrical Brownian Motion

51

conclude the proof of (2.45) here. The property (2.46) follows identically in analogy with Proposition 2.6. We make explicit that .ηB is defined by the mapping ei × ω × t → η(ω)Bt (ei , ω).

.

⨆ ⨅ Remark 2.4 Although we have not explicitly addressed the construction of the integral over a time interval .[s, t] where .s > 0, this can be done without any extra difficulty just as in the standard real valued case. If we were to just consider the integral over .[s, t] in Proposition 2.17, then the results extend to any .Fs -measurable .φ, η in Proposition 2.17. To show this we revisit Proposition 2.4, appreciating that the .Fs -measurability does not disturb the measurability requirements of the simple process. We also extend the Stochastic Dominated Convergence Theorem to this setting. H Lemma 2.7 Let .(B n ) be a sequence in .I¯ (W) such that there exist processes .B : Ω × [0, ∞) → L 2 (U; H) and .Q ∈ L2 ([0, ∞); R) .P − a.s., with the properties that for every .T > 0, .P × λ − a.e. .(ω, t) ∈ Ω × [0, T ]: ‖ ‖ 1. .‖Btn (ω)‖L 2 (U;H) ≤ |Qt (ω)| for all .n ∈ N. 2. .(Btn (ω)) is convergent to .Bt (ω) in .L 2 (U; H). H Then .B ∈ I¯ (W), and for every .t > 0, there exists a subsequence indexed by .(nk ) such that ⎰ t ⎰ t nk . lim Br dWr = Br dWr (2.47) nk →∞ 0

0

P − a.s..

.

Proof The proof is identical to that of Lemma 2.1, simply replacing .H by L 2 (U; H) when dealing with the integrands and using the appropriate Itô isometry Proposition 2.15. ⨆ ⨅

.

We now shift attentions to results regarding the martingale properties of the integral. Proposition 2.18 For .B ∈ IH (W), the Itô stochastic integral ⎰

t

B(s)dWs

.

0

belongs to .M2c (H). If .B ∈ IH (W) only, then the integral is a continuous local martingale. Proof This follows immediately from Propositions 2.7, 2.8 and, subsequently, Proposition 2.9. ⨆ ⨅

52

2 Stochastic Calculus in Infinite Dimensions

The martingality of the integral also allows us to consider the quadratic variation as defined in Definition 2.16. Proposition 2.19 For .B ∈ IH (W), we have that ⎡⎰

⎤

·

⎰

t

=

Br dWr

.

0

0

t

‖Br ‖2L 2 (U;H) dr.

(2.48)

Proof At each time t, the integral ⎰

t

Br dWr

.

0

is defined to be the .L2 (Ω; H) limit of the sequence n ⎰ ⎲

t

.

i=1

0

Br (ei )dWri .

We look to infer the quadratic variation of this sequence of processes using Proposition 2.10, to then apply Lemma 2.5. We claim that ⎡ n ⎰ ⎲

·

.

0

i=1

⎤ =

Br (ei )dWri

⎰ t⎲ n 0 i=1

t

‖Br (ei )‖2H dr,

which is to say ‖ n ⎰ ‖2 ⎰ t⎲ n ‖⎲ t ‖ ‖ i‖ ‖Br (ei )‖2H dr .‖ Br (ei )dWr ‖ − ‖ ‖ 0 0 H

i=1

(2.49)

i=1

is a martingale. For the orthonormal basis .(ak ) of .H, ‖2 ‖ n ⎰ ⎰ t⎲ n ‖ ‖⎲ t ‖ i‖ ‖Br (ei )‖2H dr .‖ Br (ei )dWr ‖ − ‖ ‖ 0 0 H

i=1

=

/ n ⎰ ∞ ⎲ ⎲ k=1

=

i=1

t 0

−

\2 −

Br (ei )dWri , ak

n ⎲ n /⎰ ∞ ⎲ ⎲ k=1 i=1 j =1

i=1

0

H t

Br (ei )dWri , ak

⎰ t⎲ n 0 i=1

‖Br (ei )‖2H dr

⎰ t⎲ n 0 i=1

\ /⎰ H

t

‖Br (ei )‖2H dr \ j

Br (ej )dWr , ak 0

H

2.5 Integration with Respect to Cylindrical Brownian Motion

=

⎛ ∞ n /⎰ ⎲⎲

t

0

k=1 i=1

⎛

+⎝

\2 Br (ei )dWri , ak

∞ ⎲ /⎰ ⎲ k=1 i/=j

=

n ⎲

t 0

H

⎰ t⎲ n

−

0 i=1

53

⎞ ‖Br (ei )‖2H dr

\ /⎰ Br (ei )dWri , ak

H

t

⎞

\ j Br (ej )dWr , ak

⎠ H

0

⎞ ⎛‖⎰ ‖2 ⎰ t ‖ ‖ t ‖ Br (ei )dW i ‖ − ‖Br (ei )‖2H dr r‖ ‖ H

0

i=1

⎛ +⎝

∞ ⎲ /⎰ ⎲ k=1 i/=j

t 0

0

\ /⎰ Br (ei )dWri , ak

H

⎞

\

t

⎠.

j

Br (ej )dWr , ak 0

H

Inspecting the last equality, by Proposition 2.10 we have that ⎞ ⎛‖⎰ ‖2 ⎰ t n ⎲ ‖ t ‖ i‖ 2 ‖ ‖Br (ei )‖H dr . ‖ Br (ei )dWr ‖ − H

0

i=1

0

is a finite sum of martingales, so the process defined in (2.49) will itself be a martingale as well if we show that the same is true for ∞ ⎲ /⎰ ⎲

t

.

0

k=1 i/=j

\ /⎰ Br (ei )dWri , ak

\

t

j

Br (ej )dWr , ak

H

0

(2.50)

. H

We consider the above at first for each fixed k, rewriting it as .

⎞ ⎛⎰ t ⎞ ⎲ ⎛⎰ t 〈 〉 j 〈Br (ei ), ak 〉H dWri . Br (ej ), ak H dWr 0

i/=j

(2.51)

0

Note that for each .i /= j , ⎛⎰ .

0

t

〈 〉 j Br (ej ), ak H dWr

⎞ ⎛⎰ 0

t

⎞ 〈Br (ei ), ak 〉H dWri

is the product of two independent real valued square integrable martingales, so is itself a martingale, and hence the finite sum given in (2.51) retains the martingale property. We emphasize again that the martingale property is always considered with respect to the fixed filtration .(Ft ) of our probability space. We wish to show that the martingale property remains true in the limit of the infinite sum for (2.50). Convergence of the infinite sum is defined .P − a.s., and it is sufficient to show that the convergence also holds in .L1 (Ω; R) as this topology retains the martingale property. For this we show that the sequence is Cauchy in

54

2 Stochastic Calculus in Infinite Dimensions

L1 (Ω; R), taking the difference of the lth and mth terms to see that

.

| | | l ⎛⎰ t ⎞ ⎛⎰ t ⎞| | ⎲ ⎲ | 〈 〉 j 〈Br (ei ), ak 〉H dWri || Br (ej ), ak H dWr .E | | 0 0 |k=m+1 i/=j | l ⎲ ⎲

≤

k=m+1 i/=j

|⎛⎰ t ⎞ ⎛⎰ t ⎞| | | 〈 〉 j 〈Br (ei ), ak 〉H dWri || Br (ej ), ak H dWr E ||

l 1 ⎲ ⎲ ≤ E 2

0

0

⎡⎛⎰ 0

k=m+1 i/=j

≤n

2

l ⎲

⎛⎰

≤n

l ⎲

0

⎰

∞ ⎲

t

sup E 0

k=m+1 i

≤ n2

t

sup E

k=m+1 i 2

⎰

t

sup E

k=m+1 i

t

0

〈 〉 j Br (ej ), ak H dWr

⎞2

⎛⎰

t

+ 0

⎞2 ⎤ 〈Br (ei ), ak 〉H dWri

⎞2 〈Br (ei ), ak 〉H dWri

〈Br (ei ), ak 〉2H dr 〈Br (ei ), ak 〉2H dr

having used the Itô isometry. This is a monotone decreasing sequence in m, convergent to zero, hence the Cauchy property is shown so there exists a limit in .L1 (Ω; R) which must agree with the .P − a.s. limit (we can take a .P − a.s. convergent subsequence from the .L1 (Ω; R) convergence) and the martingale property of the process defined ⎡⎰ · in (2.50), ⎤ and hence (2.49) is shown. By applying Lemma 2.5, we deduce that . 0 Br dWr t is the .L1 (Ω; R) limit of the sequence ⎰ t⎲ n .

0 i=1

‖Br (ei )‖2H dr

in n. Similarly to the analysis just conducted, we can show that this sequence is Cauchy in .L1 (Ω; R) and its limit agrees with the .P − a.s. limit, which is of course ⎰ .

0

t

‖Br ‖2L 2 (U;H) dr

taking the infinite sum through the integral with either Tonelli’s Theorem (identifying the infinite sum as an integral with respect to the counting measure) or the Monotone Convergence Theorem. The proof is concluded. ⨆ ⨅ We also have the analogous result to Proposition 2.11.

2.5 Integration with Respect to Cylindrical Brownian Motion

55

Proposition 2.20 Let .B ∈ IH T (W) and consider any sequence of partitions { } Il := 0 = t0l < t1l < · · · < tkl l = T

.

with .maxj |tjl − tjl −1 | → 0 as .l → ∞. Then for all .t ∈ [0, T ], for any .ε > 0, ⎫⎞ | ⎛ ⎧| | ‖ ‖2 | ⎪ ⎪ ⎰ t ⎬ | ‖ ⎨| ⎲ ‖⎰ tjl +1 ⎟ | ‖ ‖ ⎜ | 2 ‖Br ‖L 2 (U;H) dr | > ε ⎠ = 0. . lim P ⎝ | Br dWr ‖ − ‖ l ⎪ | ‖ ‖ | ⎪ l→∞ 0 ⎭ ⎩|t l ≤t tj | H j +1 (2.52) Proof Following the method used in Proposition 2.11, we would again like to reduce this to a familiar case and extrapolate the result to the limit. We introduce a sequence of stopping times .(τ n ) defined at every .n ∈ N by ⎧

⎫ ‖2 ‖⎰ t ⎰ t ‖ ‖ 2 ‖ ‖ ‖Br ‖L 2 (U;H) dr ≥ n . .τ := n ∧ inf t ∈ [0, T ] : ‖ Br dWr ‖ + n

0

H

0

For every n we define the process B·n := B· 1·≤τ n

.

and now look to show that |⎞ ⎛| | ‖ ‖2 | ⎰ t | ‖ | ⎲ ‖⎰ tjl +1 ‖ n ‖2 |⎟ ‖ ‖ ⎜| n ‖B ‖ 2 . lim E ⎝| Br dWr ‖ − dr ‖ r L (U;H) |⎠ = 0. | ‖ tl ‖ |l l→∞ 0 j | |tj +1 ≤t H This is precisely in line with the method of Proposition 2.11. We have that |⎞ ⎛| | ‖ ‖2 | ⎰ t | ‖ | ⎲ ‖⎰ tjl +1 ‖ ‖ 2 |⎟ ‖ ‖ ⎜| n n ‖B ‖ 2 .E ⎝| Br dWr ‖ − dr |⎠ ‖ r L (U;H) | ‖ tl ‖ |l 0 j | |tj +1 ≤t H |⎞ ⎛| | | ‖ ‖2 ⎰ t⎲ ∞ | ⎲ ‖⎰ tjl +1 | ‖ ‖ ‖ 2 ⎜| ‖ ‖ ‖B n (ei )‖ dr ||⎟ = E ⎝| Brn dWr ‖ − ‖ r H |⎠ |l ‖ l ‖ 0 i=1 |tj +1 ≤t tj | H |⎞ ⎛| | | / \2 ⎰ t⎲ ∞ ∞ ⎲ ∞ ⎰ tl |⎲ ⎲ 〈 n 〉2 ||⎟ j +1 ⎜| n = E ⎝| Br (ei ), ak H dr |⎠ Br dWr , ak − l |l | 0 i=1 k=1 |tj +1 ≤t k=1 tj | H

(2.53)

56

2 Stochastic Calculus in Infinite Dimensions

|⎞ ⎛| | | ⎛⎰ l ⎞2 ⎰ ∞ ∞ ∞ |⎲ ⎲ tj +1 〈 t ⎲⎲〈 〉 〉2 ||⎟ ⎜| n n Br , ak H dWr − Br (ei ), ak H dr |⎠ = E ⎝| l | |l 0 k=1 i=1 | |tj +1 ≤t k=1 tj | |⎞ ⎛ | | ⎛⎰ l ⎞2 ⎰ ∞ ∞ |⎲ ⎲ | tj +1 〈 t⎲ ‖〈 〉 〉 ‖2 ⎜| |⎟ n n ‖ B (ei ), ak ‖ 2 = E ⎝| Br , ak H dWr − dr r H L (U;R) ||⎠ l |l 0 k=1 |tj +1 ≤t k=1 tj | |⎞ ⎛| | | ⎛ ⎞2 ⎰ ∞ | | ⎲ ⎰ tjl +1 〈 t ⎲ ‖ ‖ 〈 〉 〉 2 |⎟ ⎜| ‖ B n (ei ), ak ‖ 2 ≤ Brn , ak H dWr − E ⎝| dr r H L (U;R) ||⎠ l |l 0 k=1 | |tj +1 ≤t tj having applied Proposition 2.16 and the Dominated Convergence Theorem to take the infinite sum in k through ⎰ · 〈 the 〉 time integral and expectation. From Propositions 2.16 and 2.18, then . 0 Brn , ak H dWr belongs to .M2c , with quadratic variation ‖2 ⎰t ‖ ‖ ‖ . 0 〈Br (ei ), ak 〉H L 2 (U;R) dr coming from Proposition 2.19. Just as we used in Proposition 2.11, we have that for each fixed .k ∈ N, |⎞ ⎛| | | ⎛ ⎞2 ⎰ | | ⎲ ⎰ tjl +1 〈 t ‖〈 n 〉 〉 ‖2 |⎟ ⎜| n ‖ ‖ Br (ei ), ak H L 2 (U;R) dr |⎠ = 0 Br , ak H dWr − . lim E ⎝| l | |l l→∞ 0 t j | |tj +1 ≤t so it is sufficient to justify the interchange of limit in l and summation in k. This follows identically to the justification in Proposition 2.11, appealing this time to the Itô isometry Proposition 2.15. ⨆ ⨅

Chapter 3

Stochastic Differential Equations in Infinite Dimensions

In this chapter we establish a framework for the study of stochastic partial differential equations (SPDEs), which are evolution equations involving integration of the form introduced in the previous chapter. Through this framework we define notions of solutions for an abstract SPDE, incorporating both unbounded (in the sense of differential operators) and Stratonovich noise. One main result is the rigorous conversion between Itô and Stratonovich forms under an unbounded noise.

3.1 The Stratonovich Integral We look at first to define the Stratonovich Integral with respect to a one dimensional martingale, before doing so with respect to a Cylindrical Brownian Motion. Recall that the Itô stochastic integral was constructed in Definitions 2.10, 2.20, and 2.22. ¯ 2 and .Ψ ∈ IH ∩ M ¯ 2 (H ), the Stratonovich stochastic Definition 3.1 For .M ∈ M c c M integral is defined as ⎰ .

t

⎰ Ψ s ◦ dMs :=

0

0

t

1 Ψ s dMs + [Ψ, M]t . 2

¯ 2 (H) for every .ei and the limit Definition 3.2 For .B ∈ IH (W) such that .Bei ∈ M c .

∞ ⎲ [Bei , W i ]t

(3.1)

i=1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 D. Goodair, D. Crisan, Stochastic Calculus in Infinite Dimensions and SPDEs, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-69586-5_3

57

58

3 Stochastic Differential Equations in Infinite Dimensions

is well defined in .L2 (Ω; H), the Stratonovich stochastic integral is defined as ⎰

t

.

∞ ⎛⎰ ⎲

B(s) ◦ dWs :=

0

t

0

i=1

⎞ 1 Bei (s)dWsi + [Bei , W i ]t , 2

where the limit is taken in .L2 (Ω; H). The class of such processes will be denoted H .I◦ (W). H It will be necessary to extend this definition to processes .B ∈ I¯ (W), but we encounter more technical issues in trying to construct a sequence of stopping times such that the stopped process belongs to .IH ◦ (W). We find it simplest to give the definition below.

Definition 3.3 Suppose that there exists a sequence of stopping times .(τn ) which are .P − a.s. monotone increasing and convergent to infinity such that: 1. For every n, the process B n (·) := B(·)1·≤τ n

.

belongs to .IH (W). 2. For every n and i, the process Beτin (·) := Bei (· ∧ τ n )

.

¯ 2 (H ). belongs to .M c 3. The limit ∞ ⎲ .

[Beτin , W i ]t

i=1

is well defined in .L2 (Ω; H). Then the Stratonovich stochastic integral is defined at any .t ≥ 0 as ⎰ .

0

t

B(s) ◦ dWs := lim

n→∞

⎛ ∞ ⎛⎰ ⎲ i=1

0

t

Beni (s)dWsi

1 + [Beτin , W i ]t 2

⎞⎞ ,

where the limit is taken .P − a.s. in .H, and the infinite sum in .L2 (Ω; H). The class H of such processes will be denoted .I¯ ◦ (W). This definition is of course completely analogous to the localization procedure used in the previous constructions, for example, Definition 2.22, except we postulate in the first instance the existence of the localizing sequence .(τn ). There does not

3.2 Strong Solutions in the Abstract Framework

59

appear to be a simple canonical way to construct such a sequence of localizing ¯ 2 (H ) if one assumes only local martingality of these times whereby each .Beτin ∈ M c component processes, and hence we assume the required properties explicitly.

3.2 Strong Solutions in the Abstract Framework We now establish a framework in which to formulate SPDEs, posed for a quartet of embedded Hilbert Spaces V ͨ→ H ͨ→ U ͨ→ X

.

where the embedding is meant as a continuous linear injection. Each Hilbert Space plays a role in defining solutions, and this structure of embeddings allows us to consider operators which are not bounded on a given Hilbert Space (particularly nonlinear and differential operators in applications). We introduce at first the Itô SPDE ⎰ t ⎰ t .Ψ t = Ψ 0 + QΨ s ds + GΨ s dWs , (3.2) 0

0

where .W continues to be a Cylindrical Brownian Motion over .U relative to our fixed filtered probability space .(Ω, F, (Ft ), P) with representation (2.36). We impose now some conditions on the operators relative to these spaces. To do so we define the general map .K˜ = K˜ c,p,q : H → R by ( p q ) ˜ K(φ) := c 1 + ‖φ‖U + ‖φ‖H

.

for any constants .c, p, q independent of .φ. Assumption 3.1 .Q : V → U is measurable, and there exists a map .K˜ = K˜ c,p,q such that for any .φ ∈ V , .

˜ ‖Qφ‖U ≤ K(φ)[1 + ‖φ‖2V ].

Assumption 3.2 .G is understood as a measurable operator G : U → L 2 (U; X),

.

G|H : H → L 2 (U; U ),

defined over .U by its action on the basis vectors G(·, ei ) := Gi (·).

.

G|V : V → L 2 (U; H )

60

3 Stochastic Differential Equations in Infinite Dimensions

Each .Gi is linear, and there exist constants .ci such that for all .φ ∈ V , .ψ ∈ H , η ∈ U:

.

.

‖Gi η‖X ≤ ci ‖η‖U ,

‖Gi ψ‖U ≤ ci ‖ψ‖H , ∞ ⎲

‖Gi φ‖H ≤ ci ‖φ‖V

ci2 < ∞.

i=1

Here .Gi is assumed linear for the purposes of the Stratonovich conversion in Sect. 3.4 but would not otherwise be necessary. It is worth clarifying how .G is defined over .U: Fix a .φ ∈ V and consider .G(φ, ·) : U → H (the arguments here apply for the larger spaces as well). Any .α ∈ U has the representation ∞ ⎲ .

〈α, ei 〉U ei ,

i=1

where ∞ ⎲ .

〈α, ei 〉2U < ∞.

i=1

Then G(φ, ·) : α ⍿→

∞ ⎲

.

〈α, ei 〉U Gi φ

i=1

is well defined as an element of H . This is justified by showing that the sequence of partial sums is Cauchy in H : Note that from Cauchy–Schwarz, ‖ n ‖2 ⎛ n ⎞⎛ n ⎞ ‖⎲ ‖ ⎲ ⎲ ‖ ‖ 2 2 〈α, ei 〉U Gi φ ‖ ≤ 〈α, ei 〉U ‖Gi φ‖H .‖ ‖ ‖ i=m i=m i=m H ⎛∞ ⎞⎛ ∞ ⎞ ⎲ ⎲ 2 2 2 〈α, ei 〉U ≤ ci ‖φ‖V , i=m

i=m

which approaches zero as .m → ∞ as the sums are finite. We introduce now the first notion of our strong solutions. Definition 3.4 Let .Ψ 0 : Ω → H be .F0 -measurable. A pair .(Ψ, τ ) where .τ is a .P − a.s. positive stopping time and .Ψ, is a process such that for .P − a.e. 2 1 and .Ψ (ω)1 .ω, .Ψ · (ω) ∈ C ([0, T ]; H ) · ·≤τ (ω) ∈ L ([0, T ]; V ) with .Ψ · 1·≤τ

1 We make the definition for a continuous process as this property is desirable, but it is not necessary to make the definition and we may not always have continuity. We shall see that one only requires .L∞ ([0, T ]; H ) regularity to construct the integrals in (3.2), and hence we face no issue if and when the continuity is relieved.

3.2 Strong Solutions in the Abstract Framework

61

progressively measurable in V is said to be a local strong solution of the equation (3.2) if the identity ⎰

t∧τ

Ψt = Ψ0 +

.

⎰ QΨ s ds +

0

t∧τ

GΨ s dWs

(3.3)

0

holds .P − a.s. in U for all .t ∈ [0, T ]. Remark 3.1 Note that if .(Ψ, τ ) is a local strong solution of equation (3.2), then Ψ · = Ψ ·∧τ due to the integral representation (3.3). That is to say, the solution is automatically stopped at the local time of existence. Moreover the progressive measurability condition on .Ψ · 1·≤τ may look a little suspect as .Ψ 0 itself may only belong to H and not V making it impossible for .Ψ · 1·≤τ to be even adapted in V . We have mildly abusing notation here; what we really ask is that there exists a process .Ф which is progressively measurable in V and such that .Ф· = Ψ · 1·≤τ almost surely over the product space .Ω × [0, T ] with product measure .P × λ. In particular, the processes .Ф and .Ψ may disagree at time zero. .

Let us take a few moments to process this definition and ensure that the integrals make sense with the given regularity of the solution and the operators .Q, .G. The time integral in (3.3) is well defined in U as a Bochner Integral: First of all 2 .Ψ · (ω)1·≤τ (ω) ∈ L ([0, T ]; V ) for .P − a.e. .ω so .Ψ · (ω)1·≤τ (ω) : [0, t] → V is measurable and hence .Q(Ψ · (ω)1·≤τ (ω) ) : [0, t] → U is measurable from Assumption 3.1. Moreover, the mapping .Q(Ψ · (ω)1·≤τ (ω) )1·≤τ (ω) : [0, t] → U is again measurable, and we have that ⎰ .

t∧τ (ω)

⎰

t

Q (Ψ s (ω)) ds =

0

) ( Q Ψ s (ω)1s≤τ (ω) 1s≤τ (ω) ds

0

so the required measurability in order to define the integral is satisfied. In this vein, we have that ⎰ .

0

t∧τ (ω)

⎰ ‖Q (Ψ s (ω))‖U ds ≤

t∧τ (ω) 0

≤

⎡ ⎤ K˜ (Ψ s (ω)) 1 + ‖Ψ s (ω)‖2V ds

sup s∈[0,t∧τ (ω)]

⎤⎰ ⎡ ˜ K (Ψ s (ω)) 0

t∧τ (ω)

1 + ‖Ψ s (ω)‖2V ds

0. Definition 3.7 A pair .(Ψ, Θ) such that there exists a sequence of stopping times (θj ) which are .P − a.s. monotone increasing and convergent to .Θ, whereby .(Ψ ·∧θj , θj ) is a local strong solution of the equation (3.2) for each j , is said to be a maximal strong solution of the equation (3.2) if for any other pair .(Ф, 𝚪) with this property, then .Θ ≤ 𝚪 .P − a.s. implies .Θ = 𝚪 .P − a.s.. .

Definition 3.8 A maximal strong solution .(Ψ, Θ) of the equation (3.2) is said to be unique if for any other such solution .(Ф, 𝚪), then .Θ = 𝚪 .P − a.s. and P ({ω ∈ Ω : Ψ t (ω) = Фt (ω)

.

∀t ∈ [0, T ∧ Θ(ω))}) = 1.

The remainder of this section is devoted to the following result. Theorem 3.3 Define a “local regular solution” of the equation (3.2) to be a local strong solution .(Ψ, τ ), with the alteration that the progressive measurability is either satisfied for the genuine process .Ψ · 1·≤τ in V (and thus, not a version of it) or of .Ψ in H . Suppose that there exists a local regular solution of the equation (3.2) and that any local regular solution is unique. Then there exists a unique maximal regular solution of the equation (3.2). To this end, we state and prove an important result in both this context and future ones. Proposition 3.1 Let ⋃ .(Ak ) be a partition of .Ω, which is to say that .Ak ∩ Aj = ∅ when .k /= j and . k Ak = Ω, where each .Ak ∈ F0 . Consider a sequence .(Ψ k0 ), each .Ψ k0 : Ω → H .F0 -measurable, and let .((Ψ k , τ k )) be local strong solutions of the equation (3.2) corresponding to .(Ψ k0 ). Then the pair .(Ψ, τ ) defined by Ψ=

∞ ⎲

.

Ψ k 1Ak ,

k=1

τ=

∞ ⎲

τ k 1Ak

k=1

is a local strong solution of the equation (3.2) for the initial condition Ψ0 =

∞ ⎲

.

Ψ k0 1Ak .

k=1

Proof We first note that the infinite sums in defining .(Ψ, τ ) and .Ψ 0 are given by a single term for any .ω, so the limits are trivially defined .P − a.s. in .C([0, T ]; U ), .R, and .R, respectively. As each .Ak ∈ F0 , then .1Ak is .F0 -measurable and hence each k .τ 1Ak remains a stopping time so their .P − a.s. limit .τ is again a stopping time. It is clear that for .P − a.e. .ω, .(Ψ(ω), τ (ω)) = (Ψ k (ω), τ k (ω)) for some k, and hence pathwise properties of the local strong solution are retained. That is, for .P − a.e. ∞ 2 .ω, .Ψ · (ω) ∈ L ([0, T ]; H ) and .Ψ · (ω)1·≤τ (ω) ∈ L ([0, T ]; V ). We next verify

3.3 Uniqueness and Maximality

65

the identity (3.3), where progressive measurability (thus ensuring that the stochastic integral is indeed well defined) is shown afterward. As the .(Ak ) partition .Ω, it is sufficient to show that for every .t ∈ [0, T ], the identity ⎰ 1Ak Ψ t = 1Ak Ψ 0 + 1Ak

t∧τ

.

0

⎰ QΨ s ds + 1Ak

t∧τ

GΨ s dWs

(3.5)

0

holds .P − a.s. in U . We now look to reduce each term, seeing that 1Ak Ψ t = 1Ak Ψ kt

.

⎰ 1Ak

t∧τ 0

1Ak Ψ 0 = 1Ak Ψ k0 ⎰ t∧τ k QΨ s ds = 1Ak QΨ ks ds 0

holds .P − a.s. in U , where the last equality is justified by ⎰ 1Ak

t∧τ

.

0

)2 ( QΨ s ds = 1Ak ⎰ = 1Ak

⎰ 0

t

= 1Ak

t

= 1Ak

t

0

1Ak 1s≤τ QΨ s ds

1Ak 1Ak ∩{s≤τ k } Q(Ψ ks 1Ak )ds

0

⎰

⎰ QΨ s ds = 1Ak

1Ak 1Ak ∩{s≤τ } Q(Ψ s 1Ak )ds

0

⎰

t∧τ

t∧τ k

0

Q(Ψ ks )ds

simply reversing the procedure to obtain the last line. We can apply the same arguments for the stochastic integral to obtain that ⎰ 1Ak

t∧τ

.

0

⎰ GΨ s dWs = 1Ak

0

t∧τ k

GΨ ks dWs

(3.6)

though we now have to be careful in justifying passage of .1Ak through the stochastic integral; as .1Ak is .F0 -measurable, then we can apply Proposition 2.17. Therefore, to verify (3.5), it is sufficient to demonstrate that ⎰ 1Ak Ψ kt = 1Ak Ψ k0 + 1Ak

.

0

t∧τ k

⎰ QΨ ks ds + 1Ak

0

t∧τ k

GΨ ks dWs ,

but this follows from the fact that .(Ψ k , τ k ) is a local strong solution for the initial condition .Ψ k0 . To conclude that .(Ψ, τ ) is a local strong solution for the initial condition .Ψ 0 , it only remains to show that .Ψ · 1·≤τ is progressively measurable in V , which we deduce from

66

3 Stochastic Differential Equations in Infinite Dimensions

Ψ · 1·≤τ =

⎛∞ ⎲

.

k=1

⎞ Ψ k· 1Ak

1·≤τ =

∞ ⎲

Ψ k· 1·≤τ k 1Ak ,

k=1

where the limit can be taken pointwise almost everywhere over the product space Ω×[0, T ] in V 2 (again for each fixed element of this product space, the infinite sum is in reality just a single term). From measurability of solutions, we ∑the progressive k have that for each N , the process . N k=1 Ψ · 1·≤τ k 1Ak is progressively measurable (this is undisturbed by the .F0 -measurable .1Ak ) and hence for each fixed S is measurable as a mapping .Ω×[0, S] → V where we equip .Ω×[0, S] with the sigma algebra .FS × B([0, S]). The pointwise .P × λ − a.e. limit as .N → ∞ preserves the measurability which concludes the argument that .(Ψ, τ ) is a local strong solution of the equation (3.2) for the initial condition .Ψ 0 . ⨆ ⨅

.

With this in place we set up notation for proving Theorem 3.3. We define .X as the set of all stopping times .σ such that there exists a process .Ψ for which .(Ψ, σ ) is a local regular solution. We also define .Y as the set of all stopping times given by the .P − a.s. limit of monotone increasing elements of .X . We prove the existence of a maximal solution by showing that the maximum of any two elements of .X is again in .X , a property which we use for the sequences in .X to bound sequences in .Y which then enables an application of Zorn’s Lemma to deduce a maximal element. Of course, an assumption of Theorem 3.3 is that .X is non-empty. Lemma 3.2 Under the assumptions of Theorem 3.3, for any .σ1 , σ2 ∈ X , we have that .σ1 ∨ σ2 ∈ X . Proof Our proof adapts the methodology of Proposition 3.1, but we establish the idea first. By definition of .X we have that .(Ψ 1 , σ1 ) and .(Ψ 2 , σ2 ) are local regular solutions for some processes .Ψ 1 , Ψ 2 , for the initial condition .Ψ 10 = Ψ 20 = Ψ 0 . The uniqueness of solutions establishes that .Ψ 1·∧σ1 ∧σ2 and .Ψ 2·∧σ1 ∧σ2 are indistinguishable. Our approach is to construct a new solution .Ψ, which agrees with these indistinguishable processes up until .σ1 ∧ σ2 and then extends to .σ1 as .Ψ 1 if .σ1 > σ2 and similarly for the reverse. One can construct the process .Ψ and stopping time .σ = σ1 ∨ σ2 by Ψ := Ψ 1 1σ1 ≥σ2 + Ψ 2 1σ1 0, .Ф is progressively measurable in .H, and the identity ⎰

t

Ф t = Ф0 +

.

⎰

t

A (s, Фs )ds +

0

G (s, Фs )dWs

(4.1)

0

holds .P − a.s. in .H for every .t ≥ 0. We remark that the operators satisfy the assumptions of 3.8, 3.9 for the spaces V = H = U := H and that the conclusion of this theorem is the existence of a strong solution of (4.1) in the sense of Definition 3.5, taken beyond T .

.

Proof With .H finite dimensional, we first restrict ourselves to finitely many Brownian Motions in our stochastic integral to make things classical. Fixing any k .S > 0, let us define the operator .G : [0, S] × H × U → H on the basis vectors .(ei ) k of .U by .G (·, ·, ei ) = G (·, ·, ei ) for .i ≤ k, and zero otherwise. We consider at first the equation ⎰ Фkt = Фk0 +

t

.

0

⎰ A (s, Фks )ds +

0

t

G k (s, Фks )dWs

or equivalently ⎰ Фkt = Фk0 +

.

0

t

A (s, Фks )ds +

k ⎰ ⎲ i=1

t 0

Gi (s, Фks )dWsi

for .Фk0 := Ф0 . The existence and uniqueness of solutions to this finite dimensional system is then classical (for solutions defined as in the theorem), see, e.g., [44, Theorems 5.2.5 and 5.2.9]. Consider now solutions .Фj , Фk for .j < k arbitrary, which therefore satisfy the difference equation

4.1 Existence and Uniqueness in Finite Dimensions

k .Фr

j − Фr

⎰ = 0

r

A

(s, Фks ) − A

j (s, Фs )ds

85

⎰

r

+

j

G k (s, Фks ) − G j (s, Фs )dWs

0

for any .r ∈ [0, ∞). By applying the energy identity Proposition 4.2, for the spaces all taken to be .H, we see further that the identity ⎰ ‖ ‖ ‖ k j ‖2 ‖Фr − Фr ‖ = 2

.

H

⎰

r 0

r

+

/ \ j j A (s, Фks ) − A (s, Фs ), Фks − Фs ds H

‖ ‖ ‖ k j ‖2 ‖G (s, Фks ) − G j (s, Фs )‖

L 2 (U;H)

0

⎰

r

+2 0

ds

/ \ j j G k (s, Фks ) − G j (s, Фs ), Фks − Фs dWs H

holds .P − a.s.. We use Cauchy–Schwarz to move to an inequality and rewrite the quadratic variation term to give us the bound ‖ ‖ ‖ k j ‖2 ‖Фr − Фr ‖ H ⎛ ⎞ ⎰ r ‖ j ‖ ‖ ‖ ‖ ‖2 ⎲ ‖ ‖ j ‖ ‖ j‖ j ‖ ≤ ⎝2‖A (s, Фks )−A (s, Фs )‖ ‖Фks −Фs ‖ + ‖Gi (s, Фks )−Gi (s, Фs )‖ ⎠ ds

.

H

0

⎰ +

r

H

⎰ k ‖ ‖2 ⎲ ‖ k ‖ (s, Ф ) ds + 2 ‖Gi s ‖ H

0 i=j +1

r 0

i=1

/ \ j j G k (s, Фks ) − G j (s, Фs ), Фks − Фs dWs . H

In one step now we bound the stochastic integral by its absolute value, take the supremum over all such r up to any arbitrary time .t ∈ [0, ∞), and employ the Lipschitz assumption to see that

.

⎰ t‖ ⎰ ‖ ‖ ‖ ‖ k ‖ k j ‖2 j ‖2 sup ‖Фr − Фr ‖ ≤ c ‖Фs − Фs ‖ ds + H

r∈[0,t]

H

0

|⎰ | + 2 sup || r∈[0,t]

0

r

t

k ‖ ‖2 ⎲ ‖ ‖ ‖Gi (s, Фks )‖ ds

0 i=j +1

H

| / \ | j j k k j k G (s, Фs ) − G (s, Фs ), Фs − Фs dWs || H

for a generic constant c, allowed to depend on t. We want to take the expectation here but have to be slightly careful in ensuring that the expectation is finite; to this end we consider the stopping times ‖ ‖ ‖ ‖ ‖ ‖ ‖ j‖ τR := R ∧ inf{s ≥ 0 : max{‖Фks ‖ , ‖Фs ‖ } ≥ R}

.

H

H

86

4 A Toolbox for Nonlinear SPDEs

and the process defined for any fixed R by ˜ ks := Фks 1s≤τR , Ф

˜ js := Фjs 1s≤τR . Ф

.

From the continuity of the processes .Фk , Фj , then ‖.(τR )‖ is a‖ .P − ‖ a.s. monotone ‖ ˜ k‖ ‖ ˜ j ‖ increasing sequence convergent to infinity and .max{‖Фs ‖ , ‖Фs ‖ } ≤ R for any H H .s ≥ 0. It is trivial that these processes satisfy the same inequality

.

⎰ t‖ ⎰ ‖ ‖ ‖ j ‖2 j ‖2 ‖˜k ‖˜k ˜ ˜ Ф sup ‖Ф − Ф ≤ c − Ф ds + ‖ ‖ ‖ r r s s H

r∈[0,t]

H

0

|⎰ | + 2 sup || r∈[0,t]

r

/

0

t

k ‖ ‖2 ⎲ ‖ ˜ ks )‖ ‖Gi (s, Ф ‖ ds H

0 i=j +1

k

j

k

j

˜ s ) − G j (s, Ф ˜ s ), Ф ˜s −Ф ˜s G k (s, Ф

\ H

| | dWs ||

and justify that the expectation of all terms involved is finite (indeed for the ˜ j· ˜ k· − Ф stochastic integral, using the Lipschitz assumption and the boundedness of .Ф / \ ˜ k· ) − G j (·, Ф ˜ j· ), Ф ˜ k· − Ф ˜ j· then . G k (·, Ф ∈ IR (W) so the expectation of this term H

is finite). We do now take the expectation and apply the classical Burkholder–Davis– Gundy Inequality, Theorem A.5, to give us the bound ⎰ t‖ ⎰ ‖ ‖ ‖ j ‖2 j ‖2 ‖˜k ‖˜k ˜ ˜ .E sup ‖Фr − Фr ‖ ≤ cE ‖Фs − Фs ‖ ds + E H

r∈[0,t]

H

0

⎛⎰ + cE

t

k ‖ ‖2 ⎲ ‖ ˜ ks )‖ ‖ ds ‖Gi (s, Ф

0 i=j +1

∞ / t⎲ 0 i=1

˜ ks ) − G j (s, Ф ˜ js ), Ф ˜ ks Gik (s, Ф i

H

\2 ˜ js ds −Ф H

⎞1 2

,

which we promptly reduce to ⎰ t‖ ⎰ ‖ ‖ ‖ j ‖2 j ‖2 ‖˜k ‖˜k ˜ ˜ Ф E sup ‖Ф − Ф ≤ cE − Ф ds + E ‖ s r r‖ s‖

.

H

r∈[0,t]

0

H

t

k ‖ ‖2 ⎲ ‖ ˜ ks )‖ ‖Gi (s, Ф ‖ ds

0 i=j +1

H

⎤ ⎞1 ⎛ ⎡ 2 ⎰ t ⎲ j ‖ k ‖ ‖2 ‖2 ⎲ ‖2 ‖ k j k k j ‖ ‖ ‖ ‖ ‖ ‖ ˜ ˜ ˜ ˜ ˜ + cE ⎝ ⎣ ‖Gi (s, Фs )−Gi (s, Фs )‖ + ‖Gi (s, Фs )‖ ⎦‖Фs − Фs ‖ ds⎠ . 0

i=1

H

i=j +1

H

Employing the Lipschitz assumption once more, followed by an application of Young’s Inequality, we have that ⎛ c⎝

⎰

.

0

⎞1 ⎤ 2 j ‖ k ‖ ‖ ‖ ‖ ‖ ⎲ ⎲ k j ‖2 k ‖2 k j ‖2 ‖ ‖ ‖ ˜ ˜ ˜ ˜ ˜ ⎠ ⎣ ⎦ ‖Gi (s, Фs ) − Gi (s, Фs )‖ + ‖Gi (s, Фs )‖ ‖Фs − Фs ‖ ds ⎡

t

i=1

i=j +1

H

H

4.1 Existence and Uniqueness in Finite Dimensions

87

⎤ ⎛ ⎡ ⎞1 2 ⎰ t ‖ k ‖ ‖2 ‖2 ‖ ‖2 ⎲ j‖ k ‖ j‖ ‖˜k ‖˜k ‖ ˜ ˜ ˜ ⎦ ⎠ Ф G ≤ c ⎝ ⎣‖Ф − Ф + (s, Ф ) − Ф ds ‖ ‖ ‖ ‖ ‖ i s s s s s H

0

H

i=j +1

H

⎛

⎞1 2 k ‖ ‖ ‖2 ⎰ t ‖ ‖2 ‖2 ⎲ k j k j k ‖˜ ‖ ‖˜ ‖ ‖ ‖ ˜ ˜ ˜ ⎝ ⎠ ≤ c sup ‖Фr − Фr ‖ ‖Ф s − Ф s ‖ + ‖Gi (s, Фs )‖ ds H 0

r∈[0,t]

⎛ =c

‖ ‖ ‖ ˜ k ˜ j ‖2 sup ‖Ф r − Фr ‖

H

r∈[0,t]

H

H

i=j +1

⎞1 ⎞ 1 ⎛⎰ 2 k ‖ 2 ‖ ‖ t‖ ⎲ k j ‖2 k ‖2 ‖ ‖ ˜ s −Ф ˜ s‖ + ˜ ⎠ ⎝ ‖Ф ‖Gi (s, Фs )‖ ds H

0

i=j +1

H

⎛ ⎞ ⎰ t ‖ k ‖ ‖ ‖ ‖ ‖ 2 ⎲ 2 2 2 1 c ‖ ˜ k ˜ j‖ ‖ ˜ ks − Ф ˜ js ‖ ˜ ks )‖ ⎝‖ sup ‖Ф − Фr ‖ + ≤ ‖ + ‖Gi (s, Ф ‖ ⎠ ds, ‖Ф H H H 2 r∈[0,t] r 2 0 i=j +1

and furthermore ⎰ t‖ ⎰ ‖ ‖ ‖ j ‖2 j ‖2 ‖˜k ‖˜k ˜ ˜ Ф E sup ‖Ф − Ф ≤ cE − Ф ds + cE ‖ ‖ ‖ r r s s

.

H

r∈[0,t]

H

0

t

k ‖ ‖2 ⎲ ‖ ˜ ks )‖ ‖ ds. ‖Gi (s, Ф H

0 i=j +1

(4.2) It is then a standard application of the Grönwall inequality that ⎰ ‖ ‖ j ‖2 ‖˜k ˜ .E sup ‖Фr − Фr ‖ ≤ cE H

r∈[0,t]

t

k ‖ ‖2 ⎲ ‖ ˜ ks )‖ ‖Gi (s, Ф ‖ ds,

(4.3)

H

0 i=j +1

where the new constant c incorporates .ect for the constant in (4.2). Observe also that, through very similar arguments just using the linear growth property instead of the Lipschitz one, we have that ‖ ‖2 ‖ ‖2 ‖ ˜ k‖ ‖˜ ‖ E sup ‖Ф r ‖ ≤ E‖Ф0 ‖

.

r∈[0,t]

H

H

⎞ ⎰ t⎛ ‖ k ‖ ‖ ‖ ‖ ‖ ⎲ k ‖ ‖ k‖ k ‖2 ‖ ‖ ˜ ˜ ˜ +E 2‖A (s, Фs )‖ ‖Фs ‖ + ds ‖Gi (s, Фs )‖ H

0

H

i=1

| k ⎰ | |⎲ r / | \ k k | ˜ s ), Ф ˜ s dWsi || + 2E sup | Gi (s, Ф | H 0 r∈[0,t] | i=1

H

88

4 A Toolbox for Nonlinear SPDEs

⎰ t ‖ ‖2 ‖ ‖ ‖ ‖ ‖ ‖2 ‖˜ ‖ ‖ ˜ k‖ ‖ ˜ k‖ ‖ ˜ k‖ ≤ E‖Ф0 ‖ + cE (1 + ‖Ф s ‖ )‖Фs ‖ + 1 + ‖Фs ‖ ds H

H

0

H

H

⎛⎰ t ⎛ ‖ ‖2 ⎞ ‖ ‖2 ⎞ 12 ‖ ˜ k‖ ‖ ˜ k‖ 1 + ‖Ф + cE ‖Фs ‖ ds s‖ H

0

H

‖ ‖2 ‖ ‖ ‖ ˜ k‖ ‖ ˜ k‖ to which we apply .‖Ф s ‖ ≤ 1 + ‖Фs ‖ to see that H

H

⎰ t ‖ ‖2 ‖ ‖2 ‖ ‖2 ‖ ˜ k‖ ‖˜ ‖ ‖ ˜ k‖ Ф E sup ‖Ф ≤ E + cE 1 + ‖Ф ‖ 0‖ r‖ s ‖ ds

.

H

r∈[0,t]

H

H

0

⎛⎰ t ⎛ ‖ ‖2 ⎞ ‖ ‖2 ⎞ 12 ‖ ˜ k‖ ‖ ˜ k‖ 1 + ‖Ф + cE ‖Фs ‖ ds s‖ H

0

H

and further ⎡ ‖ ‖ ⏋ ⎰ t‖ ‖ ‖ ‖2 ‖ ˜ k‖ ‖ ˜ k ‖2 ‖ ˜ ‖2 Ф E sup ‖Ф ≤ c E + 1 + cE ‖ ‖ ‖Фs ‖ ds ‖ 0 r

.

H

r∈[0,t]

H

0

H

as above, integrating the 1 and adding it as a constant. Thus we have ⎡ ‖ ‖ ⏋ ‖ ‖2 ⎡ ⏋ ‖ ˜ k‖ ‖ ˜ ‖2 2 ‖ Ф E sup ‖Ф ≤ c E + 1 = c E‖Ф + 1 , ‖ ‖ ‖ 0 0 H r

.

H

r∈[0,t]

H

which is a bound uniform in k and independent of .τR . Moreover for each fixed k we appreciate that the sequence of random variables .

‖ ‖2 ‖ ˜ k‖ sup ‖Ф r‖

r∈[0,t]

H

‖ ‖2 is monotone increasing (indexed by R) and convergent to .supr∈[0,t] ‖Фkr ‖H , .P − a.s.. Thus we may apply the Monotone Convergence Theorem to this sequence of random variables to see that ‖ ‖2 ‖ ‖2 ⎡ ⏋ ‖ ‖ ‖ ˜ k‖ 2 E sup ‖Фkr ‖ = lim E sup ‖Ф r ‖ ≤ c E‖Ф0 ‖H + 1 .

.

r∈[0,t]

H

R→∞

r∈[0,t]

H

With this bound established we can revert back to (4.3), combining with the boundedness of the .Gi to deduce that ⎰ E

.

t

k ‖ ‖2 ⎲ ‖ ‖ ‖Gi (s, Фks )‖ ds < ∞

0 i=j +1

H

4.1 Existence and Uniqueness in Finite Dimensions

89

and clearly ⎰

⎰ k ‖ ‖2 ⎲ ‖ ˜ ks )‖ ‖ ds ≤ E ‖Gi (s, Ф

t

E

.

H

0 i=j +1

t

k ‖ ‖2 ⎲ ‖ ‖ ‖Gi (s, Фks )‖ ds. H

0 i=j +1

So we can update (4.3) with the bound ⎰ ‖ ‖ j ‖2 ‖˜k ˜ E sup ‖Ф − Ф ≤ cE r r‖

.

H

r∈[0,t]

t

k ‖ ‖2 ⎲ ‖ ‖ ‖Gi (s, Фks )‖ ds H

0 i=j +1

to which we apply the same monotone convergence argument to deduce that ⎰ ‖ ‖ ‖ k j ‖2 .E sup ‖Фr − Фr ‖ ≤ cE H

r∈[0,t]

t

k ‖ ‖2 ⎲ ‖ ‖ ‖Gi (s, Фks )‖ ds.

(4.4)

H

0 i=j +1

Moreover ⎰ .

sup E k>j

t

k ‖ ‖2 ⎲ ‖ ‖ ‖Gi (s, Фks )‖ ds ≤ t sup E sup H

0 i=j +1

k>j

k ‖ ‖2 ⎲ ‖ ‖ ‖Gi (r, Фkr )‖ k ⎲

≤ t sup E sup k>j

≤ c sup

r∈[0,t] i=j +1 k ⎲

k>j i=j +1

≤c

∞ ⎲ i=j +1

≤c

H

r∈[0,t] i=j +1

∞ ⎲ i=j +1

‖ ‖2 ‖ ‖ Ct ci (1 + ‖Фkr ‖ ) H

‖ ‖2 ‖ ‖ ci E sup (1 + ‖Фkr ‖ ) r∈[0,t]

H

‖ ‖2 ‖ ‖ ci sup E sup (1 + ‖Фkr ‖ ) k>j

r∈[0,t]

H

⎛ ⎡ ⏋⎞ ci c E‖Ф0 ‖2H + 1 ,

which is a sequence in j monotone decreasing to zero. Note that we have absorbed a time dependence into the constant c, which is not meaningful. Thus in view of (4.4), ⎡

.

‖ ‖ ‖ j ‖2 lim sup E sup ‖Фkr − Фr ‖

j →∞ k>j

r∈[0,t]

H

⏋ = 0,

90

4 A Toolbox for Nonlinear SPDEs

so the sequence .(Фk ) is Cauchy in .L2 (Ω; C ([0, t]; H)) and as such we can deduce the existence) of a .Ф such that .Фk → Ф in this space (and hence, in ( 2 Ω; L2 ([0, t]; H) ) for every .t ∈ [0, S], and thus .Ф is also the .P − a.s. .L limit of a subsequence of the .(Фk ) in .C ([0, S]; H). This limit process inherits the progressive measurability (indeed it is adapted and has continuous paths in .H). It simply remains to show that .Ф satisfies the identity (4.1), so we first consider the .P − a.s. convergent subsequence .(Фkl ). Looking at the stochastic integral and employing Proposition 2.15, we have that ‖2 ‖⎰ t ⎰ t ‖ ‖ kl kl ‖ .E G (s, Фs )dWs ‖ ‖ ‖ G (s, Фs )dWs − 0

H

0

‖2 ‖⎰ t ‖ ‖ kl kl ‖ = E‖ G (s, Ф ) − G (s, Ф )dW s s‖ s ‖

H

0

⎰ t⎲ ∞ ‖ ‖2 ‖ ‖ k =E ‖Gi (s, Фs ) − Gi l (s, Фks l )‖ ds H

0 i=1

⎰

⎛ ⎞ kl ‖ ∞ ‖2 ⎲ ⎲ ‖ ‖ ⎝ ‖Gi (s, Фs )‖2H ⎠ ds ‖Gi (s, Фs ) − Gi (s, Фks l )‖ ds +

t

=E 0

≤

kl ⎲

⎰

t

E 0

i=1

≤ cE

H

i=1

i=kl +1

⎰ t ∞ ‖ ‖2 ⎲ ‖ ‖ ci ‖Фs − Фks l ‖ ds + E ci ‖Фs ‖2H ds H

0

i=kl +1

⎰ t‖ ⎰ t ∞ ‖2 ⎲ ‖ ‖ E ci ‖Фs ‖2H ds. ‖Фs − Фks l ‖ ds + H

0

0

i=kl +1

( ) So from the known .L2 Ω; L2 ([0, t]; H) convergence, we have that ⎰ .

lim

kl →∞ 0

t

⎰ G kl (s, Фks l )dWs =

t

G (s, Фs )dWs

0

with the limit taken in .L2 (Ω; H). We can thus extract a further subsequence which we denote .(Фkm ) such that this limit holds .P − a.s. in .H and is of course still such that .(Фkm ) → Ф .P − a.s. in .L2 ([0, t]; H). Therefore ‖⎰ t ‖2 ⎰ t ⎰ t‖ ‖2 ‖ ‖ ‖ km km ‖ ‖ ‖ A (s, Ф . A (s, Ф )ds − A (s, Ф )ds ≤ t ) − A (s, Ф ) ‖ ‖ ds s s s s ‖ ‖ H 0 0 0 H ⎰ t‖ ‖2 ‖ ‖ ≤ ct ‖Фs − Фks m ‖ ds 0

H

4.1 Existence and Uniqueness in Finite Dimensions

91

and so ⎰ .

⎰

t

A

lim

km →∞ 0

(s, Фks m )ds

t

=

A (s, Фs )ds

0

with the limit .P − a.s. in .H. Thus by taking the .P − a.s. limit in .H of the identity satisfied by .Фkm , we reach (4.1) as required. ⨆ ⨅ Theorem 4.2 Suppose .Ψ is another strong solution of (4.1). Then for every .S ≥ 0, P ({ω ∈ Ω : Фt (ω) = Ψ t (ω) ∀t ∈ [0, S]}) = 1.

.

Proof The method of proof here is entirely contained in that for the existence proof. Indeed we look at the energy equality satisfied by the difference of the solutions, which is ⎰ r 2 〈A (s, Фs ) − A (s, Ψ s ), Фs − Ψ s 〉H ds .‖Фr − Ψ r ‖ = 2 H 0

⎰

r

+ 0

‖G (s, Фs ) − G (s, Ψ s )‖2L 2 (U;H) ds

⎰

r

+2 0

〈G (s, Фs ) − G (s, Ψ s ), Фs − Ψ s 〉H dWs .

Following along the proof, we introduce τR := R ∧ inf{s ≥ 0 : max{‖Фs ‖H , ‖Ψ s ‖H } ≥ R}

.

and the process defined for any fixed R by ˜ s := Фs 1s≤τR , Ф

.

˜ s := Ψ s 1s≤τR . Ψ

In this case we have the inequality .

⎰ t‖ ‖ ‖2 ‖2 ‖˜ ‖˜ ˜ r‖ ˜ s‖ sup ‖Ф ‖ ≤c ‖Ф s − Ψ ‖ ds r −Ψ

r∈[0,t]

H

H

0

|⎰ | + 2 sup || r∈[0,t]

r/

˜ s ) − G (s, Ψ ˜ s ), Ф ˜s −Ψ ˜s G (s, Ф

0

\ H

| | dWs ||

‖ ‖2 ∑ ‖ ˜ ks )‖ so following all of the same steps, simply now without the . ki=j +1 ‖Gi (s, Ф ‖ H term, we deduce again that ‖ ‖2 ‖˜ ˜ r‖ E sup ‖Ф ‖ ≤0 r −Ψ

.

r∈[0,t]

H

92

4 A Toolbox for Nonlinear SPDEs

in analogy with (4.3). By the same monotone convergence argument, we have that E sup ‖Фr − Ψ r ‖2H = 0,

.

r∈[0,t]

⨆ ⨅

which gives the result.

4.2 Tightness Criteria Unsurprisingly, one route into the relative compactness methods of PDE theory is through tightness, owing to Prokhorov’s Theorem. Poignantly we can connect this weak limit of measures with a genuine process through Skorohod’s Representation Theorem, see, for example, [7, pp. 70]. Simple criteria through which we can establish tightness in the space of solutions to SPDEs will prove useful. Recalling our notions of solution, for example, Definitions 3.5 and 3.9, we consider tightness criteria in both the spaces .L2 ([0, T ]; H) and .D ([0, T ]; H) for suitably chosen .H (recall the definition of .D ([0, T ]; H) from Sect. 1.2). We note, for example, [7, pp. 124], that the Skorohod Topology is equivalent to the uniform topology when restricted to continuous functions. It is necessary to use due to the separability of the associated metric space, allowing us to invoke Prokhorov’s Theorem. This process is reviewed at the end of the section. We first give two results for tightness in .D ([0, T ]; H). This method was applied in [59] but was not established into a result, so we give a full proof here. Lemma 4.1 Let .Y be a reflexive Banach Space and .H a Hilbert Space such that .Y is compactly embedded into .H, and consider the induced Gelfand Triple Y ͨ− → H ͨ− → Y∗ .

.

For some fixed .T > 0 let .Ψ n : Ω → C ([0, T ]; H) be a sequence of measurable processes such that for every .t ∈ [0, T ], ⎛ .

sup E n∈N

‖ ‖ sup ‖Ψ nt ‖H

t∈[0,T ]

⎞ 0, y ∈ Y,

.

.

lim sup P

δ→0+ n∈N

|/ }⎞ ⎛{ \ | | | ω ∈ Ω : | Ψ n(γn +δ)∧T − Ψ nγn , y | > ε = 0. H

(4.6)

Then the( sequence )of the laws of .(Ψ n ) is tight in the space of probability measures over .D [0, T ]; Y∗ .

4.2 Tightness Criteria

93

Proof We essentially combine the tightness criteria of Theorems A.6 and A.7, in the specific case outlined ( here. )∗ First in reference to Theorem A.7 we may take E to be .Y∗ and .F to be . Y∗ , which is well known to separate points in .Y∗ from a corollary of the Hahn–Banach Theorem which asserts that for every .φ ∈ Y∗ ( ∗ )∗ there exists a .ψ ∈ Y such that .〈φ, ψ〉Y∗ ×(Y∗ )∗ = ‖φ‖Y∗ . We also note that A.7 condition 2 in Theorem ) is satisfied for .(μn ) taken to be the sequence of laws ( of .(Ψ n ) over .D [0, T ]; Y∗ , owing to the property (4.5). Indeed as .Y is compactly embedded into .H, then .H is compactly embedded into .Y∗ , so one only needs to take a bounded subset of .H for this condition. Considering the closed ball of radius M in .H, .B˜ M , we have that ⎞}⎞ ⎛{ ⎞}⎞ ⎛{ ⎛ ⎛ ≤ P ω ∈ Ω : Ψ n (ω) ∈ .P ω ∈ Ω : Ψ n (ω) ∈ / D [0, T ]; B˜ M / C [0, T ]; B˜ M ⎛⎧ ⎫⎞ ‖ n ‖ ≤P ω ∈ Ω : sup ‖Ψ (ω)‖ > M t

t∈[0,T ]

≤

‖ ‖ 1 E( sup ‖Ψ nt ‖H ) M t∈[0,T ]

≤

‖ ‖ 1 sup E( sup ‖Ψ nt ‖H ) M n∈N t∈[0,T ]

H

from which we see an arbitrarily large choice of M will justify condition 2 of Theorem remains to show that for every )∗ By applying the theorem, it only ( ∗A.7. .ψ ∈ Y the sequence of the laws of .〈Ψ n , ψ〉Y∗ ×(Y∗ )∗ is tight in the space of probability measures over .D ([0, T ]; R). By the reflexivity of .Y for every ( ∗ )∗ .ψ ∈ Y , there exists a .y ∈ Y such that .〈Ψ n , ψ〉Y∗ ×(Y∗ )∗ = 〈Ψ n , y〉Y∗ ×Y , and as .Ψ nt ∈ H .P − a.s., then this is furthermore just .〈Ψ n , y〉H . The problem is now reduced to showing tightness in .D ([0, T ]; R), which by Theorem A.6 is satisfied if we can show that for any sequence of stopping times .(γn ), .γn : Ω → [0, T ], and constants .(δn ), .δn ≥ 0, and .δn → 0 as .n → ∞: 〈 〉 1. For every .t ∈ [0, T ], the sequence of the laws of . Ψ nt , y H is tight in the space of probability measures over .R.⎛{ |/ }⎞ \ | | | 2. For every .ε > 0, .limn→∞ P ω ∈ Ω : | Ψ n(γn +δn )∧T − Ψ nγn , y | > ε = 0. H

We address each item in turn: As for 1, we are required to show that for every .ε > 0 and .t ∈ [0, T ], there exists a compact .Kε ⊂ R such that for every .n ∈ N, P

.

({

}) 〈 〉 ω ∈ Ω : Ψ nt (ω), y H ∈ / Kε < ε.

To this end define .BM as the closed ball of radius M in .R, and then P

.

|〈 }) }) ({ ({ 〈 〉 〉 | ω ∈ Ω : Ψ nt (ω), y H ∈ / BM = P ω ∈ Ω : | Ψ nt (ω), y H | > M

94

4 A Toolbox for Nonlinear SPDEs

|〈 〉 | 1 E(| Ψ nt , y H |) M (‖ ‖ ) ‖y‖H ≤ sup E ‖Ψ nt ‖H M n∈N

≤

so setting M :=

.

(‖ ‖ ) 2‖y‖H supn∈N E ‖Ψ nt ‖H ε

justifies item 1. As for 2, note that for each fixed .j ∈ N we have that .

|/ |/ \ | \ | | j | | | | Ψ (γj +δj )∧T − Ψ jγj , y | ≤ sup | Ψ n(γn +δj )∧T − Ψ nγn , y | H

H

n∈N

so in particular .

lim P

j →∞

}⎞ ⎛{|/ \ | | j | | Ψ (γj +δj )∧T − Ψ jγj , y | > ε H

}⎞ ⎛{|/ \ | | | ≤ lim sup P | Ψ n(γn +δj )∧T − Ψ nγn , y | > ε . j →∞ n∈N

H

As .(δj ) was an arbitrary sequence of nonnegative constants approaching zero, we can generically take .δ → 0+ and 2 is implied by (4.6). The proof is complete. ⨆ ⨅ We do not need to rely on the Gelfand Triple structure to obtain such a criterion. Lemma 4.2 Let .H1 , H2 be Hilbert spaces with .H1 compactly embedded into .H2 , and V any dense set in .H2 . For some fixed .T > 0, let .Ψ n : Ω → C ([0, T ]; H1 ) be a sequence of measurable processes such that ⎛ .

sup E n∈N

‖ ‖ sup ‖Ψ nt ‖H

t∈[0,T ]

⎞ 1

0,

.

δ→0+ n∈N

| ⎫⎞ | |>ε = 0. H2 |

(4.8)

Then the sequence of the laws of .(Ψ n ) is tight in the space of probability measures over .D ([0, T ]; H2 ). Proof The proof is mechanically nearly identical to that of Lemma 4.1, so we highlight only the slight technical differences. In reference to Theorem A.7 we may take E to be .H2 and .F to be the collection of functions defined for each

4.2 Tightness Criteria

95

v ∈ V by .〈·, v〉H2 , which separates points in .H2 from the density of V . We also note that condition 2 of Theorem A.7 is satisfied for .(μn ) taken to be the sequence of laws of .(Ψ n ) over .D ([0, T ]; H2 ), owing to the property (4.7). Indeed as .H1 is compactly embedded into .H2 , one only needs to take a bounded subset of .H1 ; hence considering the closed ball of radius M in .H1 , .B˜ M , we have that

.

P

.

⎞}⎞ ⎛{ ⎛ ‖ ‖ 1 sup E( sup ‖Ψ nt ‖H ) ≤ ω ∈ Ω : Ψ n (ω) ∈ / D [0, T ]; B˜ M 1 M n∈N t∈[0,T ]

precisely as in Lemma 4.1, from which we see an arbitrarily large choice of M will justify (3.3). Therefore by applying Theorem A.7 it only remains to show that for every .v ∈ V the sequence of the laws of .〈Ψ n , v〉H2 is tight in the space of probability measures over .D ([0, T ]; R). The remainder of the proof now follows exactly as in Lemma 4.1. ⨆ ⨅ For completeness, we state a criterion from the literature for tightness in the L2 ([0, T ]; H) space as well. A brief proof is given here, which is taken from [59, Lemma 5.2].

.

Lemma 4.3 Let .H1 , H2 be Hilbert Spaces such that .H1 is compactly embedded into .H2 , and for some fixed .T > 0 let .(Ψ n ) : Ω × [0, T ] → H1 be a sequence of measurable processes such that ⎰ .

T

sup E 0

n∈N

‖ n ‖2 ‖Ψ ‖ ds < ∞, s H

(4.9)

1

and for any .ε > 0, .

⎛⎧ ⎰ lim sup P ω ∈ Ω :

δ→0+ n∈N

T −δ

‖ ‖ n ‖Ψ (ω) − Ψ n (ω)‖2 ds > ε s s+δ H

⎫⎞

2

0

= 0.

(4.10)

Then the sequence of the laws of .(Ψ n ) is tight in the space of probability measures over .L2 ([0, T ]; H2 ). Proof Observe that from Chebyshev’s inequality and condition (4.9), ⎛⎧ ⎰ lim sup P ω ∈ Ω :

.

R→∞ n∈N

1 R→∞ R

≤ lim

⎛

0

⎰

T

sup E 0

n∈N

T

‖ n ‖2 ‖Ψ ‖ ds > R s H

⎫⎞

1

‖ n ‖2 ‖Ψ ‖ ds s H 1

⎞ =0

so in particular for any given .ε > 0, there exists an R such that .

sup P n∈N

⎛⎧ ⎰ ω∈Ω:

T 0

‖ n ‖2 ‖Ψ ‖ ds > R s H 1

⎫⎞
0 such that ⎛⎧ ⎰ . sup P ω∈Ω:

T −δk 0

n∈N

‖2 n ‖ s+δk (ω) − Ψ s (ω) H2 ds

‖ n ‖Ψ

1 > k

⎫⎞ ≤

ε 2k+1

(4.12)

.

Defining ⎧ ⎰ 𝚪k := φ ∈ L2 ([0, T ]; H2 ) :

T −δk

.

0

‖ ‖ ‖φs+δ (ω) − φs (ω)‖2 ds ≤ 1 k H2 k

⎫

it follows that the set A := BR ∩

∞ ⋂

.

𝚪k

k=1

is shown to be relatively compact in .L2 ([0, T ]; H2 ), and .

∞ ∞ ) ⎲ ) ε ⎲ ( ) ( ( ε sup P Ψ n ∈ / A ≤ sup P Ψ n ∈ / BR + sup P Ψ n ∈ / 𝚪k ≤ + ≤ ε, k+1 2 2 n∈N n∈N n∈N k=1

which concludes the proof.

k=1

⨆ ⨅

For clarity, we lay out some of the steps that one may take to use this theory in order to deduce the existence of solutions. We refer to strong solutions of the equation (3.2), defined in Definition 3.4: • Consider a Galerkin Approximation with solutions .(Ψ n ) existing at least locally in V , as described in Sect. 4.1 and applying Theorem 4.1. • Use the tightness criteria to show that the .(Ψ n ) are tight in the spaces 2 .D ([0, T ]; H ) and .L ([0, T ]; V ), hence in their intersection endowed with the sum metric, which is again a complete separable metric space. • Apply Prokhorov’s Theorem to deduce that the sequence of the laws of .(Ψ n ) is relatively compact in the space of probability measures over .D ([0, T ]; H ) ∩ L2 ([0, T ]; V ) endowed with the topology of weak convergence. We remark again that the space .D ([0, T ]; H ) is separable, while .C ([0, T ]; H ) is not, hence why we have appealed to the Skorohod Space. • Apply Skorohod’s Representation Theorem to deduce the existence of a new probability space, a sequence of .D ([0, T ]; H ) ∩ L2 ([0, T ]; V ) valued random variables on this new probability space with the same laws as a subsequence ˜ admitted as an almost sure limit on this of the .(Ψ n ), and a random variable .Ψ space. As the restriction of the .D ([0, T ]; H ) topology to continuous functions is ˜ is continuous in H . equivalent to the uniform topology, then this limit .Ψ

4.3 Cauchy Criteria

97

˜ is a strong solution of • Use the powerful almost sure convergence to deduce that .Ψ the equation (3.2) posed on the new probability space; this is a probabilistically weak or martingale solution. • Prove the uniqueness of solutions on this space, and apply a Yamada–Watanabe type argument (e.g., [58]) to obtain the existence of a strong solution on the original probability space. Of course, when dealing with highly nontrivial SPDEs, the approach has to be fine-tuned with potential truncations and appeals to alternate spaces. This is simply a sketch of the rough idea; one can see a full, concrete approach in the related paper [36].

4.3 Cauchy Criteria A more direct way to deduce the existence of a limiting process from the Galerkin Approximations comes from the Cauchy Property. In practice, due to potential nonlinearities, one requires some truncation to get sufficient control on the approximating sequence. More precisely for the .nth term of the sequence, one must work up to a first hitting time of this process, giving a stopping time .τn . The question is then whether we can deduce a limiting process up to some time .τ , where .0 < τ ≤ τn for all n. Such a result was proven by Glatt-Holtz and Ziane in [32, Lemma 5.1]. The result presented here is an extension of this and has important applications in the deduction of maximal and global solutions. The result of Glatt-Holtz and Ziane asserts that, under assumptions of a Cauchy property of the sequence of processes up until their first hitting times and some weak equicontinuity at time zero, then a limiting process and positive stopping time exist (which are then argued to be a local strong solution, as a limit of the Galerkin Approximation). No characterization of this stopping time is given though; hence completely separate arguments are required to consider what interval the solution exists upon. We demonstrate that if instead one imposes that the processes satisfy a weak equicontinuity assumption at all times, then the limiting stopping time can be taken as a first hitting time of the limiting process for an arbitrarily large hitting parameter. Application of this result immediately yields that solutions exist up until they blow up, removing the need for further analysis toward the interval on which solutions exist. Proposition 4.1 Fix .T > 0. For .t ∈ [0, T ] let .Xt denote a Banach space with norm .‖·‖X,t such that for all .s > t, .Xs ͨ− → Xt and .‖·‖X,t ≤ ‖·‖X,s . Suppose that n n n .(Ψ ) is a sequence of processes .Ψ : Ω |→ XT , .‖Ψ ‖X,· is adapted and .P − a.s. n 2 continuous, .Ψ ∈ L (Ω; XT ), and such that .supn ‖Ψ n ‖X,0 ∈ L∞ (Ω; R). For any given .M > 1, define the stopping times { ‖ ‖2 ‖ ‖2 } τnM,T := T ∧ inf s ≥ 0 : ‖Ψ n ‖X,s ≥ M + ‖Ψ n ‖X,0 .

.

(4.13)

98

4 A Toolbox for Nonlinear SPDEs

Furthermore, suppose .

⏋ ⎡‖ ‖2 lim sup E ‖Ψ n − Ψ m ‖X,τ M,T ∧τ M,T = 0

m→∞ n≥m

m

n

(4.14)

and that for any stopping time .γ and a sequence of stopping times .(δj ) that converge to 0 .P − a.s., ⎞ ⎛‖ ‖ ‖ n ‖2 n 2 = 0. . lim sup E ‖Ψ ‖ (4.15) M,T − ‖Ψ ‖ M,T X,(γ +δj )∧τn

j →∞ n∈N

X,γ ∧τn

M,T Then there exists a stopping time .τ∞ , a process .Ψ : Ω |→ Xτ M,T whereby ∞ .‖Ψ‖ M,T is adapted and .P − a.s. continuous, and a subsequence indexed by X,·∧τ∞ .(mj ) such that: M,T • .τ∞ ≤ τmM,T .P − a.s.. j • .limj →∞ ‖Ψ − Ψ mj ‖X,τ M,T = 0 .P − a.s.. ∞

Moreover for any .R > 0 we can choose M to be such that the stopping time { τ R,T := T ∧ inf s ≥ 0 : ‖Ψ‖2

.

M,T X,s∧τ∞

≥R

} (4.16)

{ } M,T R,T is simply .T ∧ inf s ≥ 0 : ‖Ψ‖2 ≥ R . satisfies .τ R,T ≤ τ∞ .P − a.s.. Thus .τ X,s Remark 4.1 A consequence of the properties that .supn ‖Ψ n ‖X,0 ∈ L∞ (Ω; R) and m ∞ (Ω; R). Therefore .limj →∞ ‖Ψ − Ψ j ‖ M,T = 0 .P − a.s. is that .‖Ψ‖X,0 ∈ L ‖ ‖ X,τ∞ R,T for .R > ‖‖Ψ‖X,0 ‖L∞ (Ω;R) , we have that .τ is .P − a.s. positive; hence so too is M,T τ∞ for appropriately chosen M.

.

Proof Property (4.14) implies that for any given .j ∈ N we can choose an .nj ∈ N such that for all .k ≥ nj , ⎛ ‖2 ‖ ‖ k n ‖ .E ‖Ψ − Ψ j ‖

⎞ ≤ 2−4j .

∧τkM,t X,τnM,t j

(4.17)

We shall make use of delicate manipulations of the subsequence indexed by .(nj ), and for this we introduce a new sequence of stopping times. We now impose that ‖ ‖ ‖ ‖ n ‖2 ‖ ‖ ‖ ‖ ‖ .M > 2 + sup Ψ ‖ X,0 ‖ n∈N

and define

L∞ (Ω;R)

4.3 Cauchy Criteria

99

‖ ‖ ‖ ‖ M − ‖supn∈N ‖Ψ n ‖2X,0 ‖

L∞ (Ω;R)

M˜ 2 :=

.

2

> 1.

The purpose of this is to define { ‖ ‖ ‖ ‖ } σjM := T ∧ inf s > 0 : ‖Ψ nj ‖X,s ≥ (M˜ − 1 + 2−j ) + ‖Ψ nj ‖X,0

.

and ensure that .σjM ≤ τnM,T at every .ω. Note the key difference in not squaring the j norm, and also that .M˜ > 1 so each .σjM is necessarily positive. To demonstrate the inequality, it is sufficient to show that, .P − a.s., .

⎛ ‖ ‖ ‖ ⎞2 ‖2 (M˜ − 1 + 2−j ) + ‖Ψ nj ‖X,0 ≤ M + ‖Ψ nj ‖X,0 ,

(4.18)

or more easily .

⎛ ‖ ‖ ‖ ⎞2 ‖2 M˜ + ‖Ψ nj ‖X,0 ≤ M + ‖Ψ nj ‖X,0 .

This is possible as ⎛ .

‖ ‖ ‖ ‖ ‖ ‖ ⎞2 ‖2 ‖ n‖2 ‖ ‖Ψ ‖ ‖ sup M˜ + ‖Ψ nj‖X,0 ≤ 2M˜ 2 +2‖Ψ nj‖X,0 = M − ‖ ‖ X,0 ‖

L∞ (Ω;R)

n

‖ ‖2 ≤ M + ‖Ψ nj ‖X,0 .

‖ ‖2 +2‖Ψ nj‖X,0

The property (4.18) is thus verified, so .σjM ≤ τnM,T , and hence, the subsequence j n j .(Ψ ) enjoys the same properties up until the corresponding .σjM . In particular from (4.17), ⎞ ⎡ ⎛ ⎞⏋ 1 ⎛ 2 ‖ ‖2 ‖ ‖ ≤ E ‖Ψ nj +1 − Ψ nj ‖X,σ M ∧σ M ≤ 2−2j , E ‖Ψ nj +1 − Ψ nj ‖X,σ M ∧σ M

.

j +1

j

j

j +1

(4.19) and hence in defining the sets ⎧ ‖ ‖ Ωj := ω ∈ Ω : ‖Ψ nj +1 (ω) − Ψ nj (ω)‖X,σ M (ω)∧σ M

.

j +1 (ω)

j

⎫ < 2−(j +2) ,

we have, by Chebyshev’s Inequality and (4.19), ⎛ P

.

ΩC j

⎞

j +2

≤2

We have, therefore, that

⎞ ⎛ ‖ ‖ n nj ‖ j +1 ‖ ≤ 2−j +2 . E Ψ − Ψ X,σ M ∧σ M j

j +1

(4.20)

100

4 A Toolbox for Nonlinear SPDEs ∞ ⎲ .

⎛ ⎞ P ΩC j 0 : ‖Ψ nj ‖X,s ≥ (M˜ − 1 + 2−j ) + ‖Ψ nj ‖X,0

.

so by the continuity of .‖Ψ nj ‖X,· , ‖ ‖ n ‖ ‖ ‖Ψ j ‖ M = (M˜ − 1 + 2−j ) + ‖Ψ nj ‖ . X,σ X,0

.

j

Using the definition of .Ωj , (4.20), for .j ≥ K, we have that

(4.22)

4.3 Cauchy Criteria

‖ n ‖ ‖Ψ j ‖

.

X,σjM ∧σjM+1

101

‖ ‖ ‖ ‖ − ‖Ψ nj +1 ‖X,σ M ∧σ M ≤ ‖Ψ nj +1 − Ψ nj ‖X,σ M ∧σ M < 2−(j +2) j +1

j

j

j +1

(4.23)

and also ‖ ‖ ‖ n ‖ ‖ ‖ ‖Ψ j +1 ‖ − ‖Ψ nj ‖ ≤ ‖Ψ nj +1 − Ψ nj ‖X,0 < 2−(j +2) . X,0 X,0

.

(4.24)

Combining (4.22), (4.23), and (4.24), while using that .σjM < σjM+1 , we see that ‖ n ‖ ‖Ψ j +1 ‖

.

X,σjM ∧σjM+1

‖ ‖ > ‖Ψ nj ‖X,σ M ∧σ M − 2−(j +2) j +1

j

‖ ‖ = ‖Ψ nj ‖X,σ M − 2−(j +2) j

‖ ‖ = ‖Ψ nj ‖X,0 + (M˜ − 1 + 2−j ) − 2−(j +2) ‖ ‖ > ‖Ψ nj +1 ‖X,0 − 2−(j +2) + (M˜ − 1 + 2−j ) − 2−(j +2) ‖ ‖ = ‖Ψ nj +1 ‖X,0 + (M˜ − 1 + 2−(j +1) ), (4.25) where in the last line we have used .

− 2−(j +2) − 2−(j +2) + 2−j = 2−(j +1) .

The hard work is done in showing that the set .𝚪 defined in (4.21) is empty, as on this set note that ‖ n ‖ ‖Ψ j +1 ‖

.

X,σjM ∧σjM+1

‖ ‖ ‖ ‖ ≤ ‖Ψ nj +1 ‖X,σ M ≤ ‖Ψ nj +1 ‖X,0 + (M˜ − 1 + 2−(j +1) ), j +1

ˆ K , and furtherwhich contradicts (4.25), hence .𝚪 must be empty. Thus on every .Ω M ˆ more the whole of .Ω, the sequence .(σj ) is eventually monotone decreasing (and M M as the pointwise limit .lim bounded below by 0). Furthermore we define .σ∞ j →∞ σj ˆ which must itself be a stopping time as the .P − a.s. limit of stopping times. on .Ω, M,T As mentioned, this shall prove to be our .τ∞ , and for the existence of .Ψ, we show n ˆ the subsequence .(Ψ j ) is Cauchy in .Xσ M . Every .ω ∈ Ω ˆK ˆ belongs to .Ω that on .Ω ∞

ˆ L for all .L > K. We fix arbitrary .ω ∈ Ω ˆ and for some K, and furthermore to .Ω select an associated K. At this .ω, for any .j > k ≥ K, observe that ‖ ‖ n ‖ ‖ ‖Ψ j − Ψ nk ‖ M = ‖Ψ nj − Ψ nk+1 + Ψ nk+1 − Ψ nk ‖ M X,σ∞ X,σ∞ ‖ n ‖ n ‖ ‖ n ≤ ‖Ψ j − Ψ k+1 ‖X,σ M + ‖Ψ k+1 − Ψ nk ‖X,σ M ∞ ∞ ‖ n ‖ nk+1 ‖ −(k+2) j ‖ ≤ Ψ −Ψ +2 X,σ M

.

∞

102

4 A Toolbox for Nonlinear SPDEs

≤

j ⎲

2−(l+2)

l=k

≤ 2−(k+1) having carried out an inductive argument in the penultimate step. We are thus free to take K large enough so that this difference is arbitrarily small; therefore there exists a limit in the Banach Space .Xσ∞ M , which we call .Ψ. The process .‖Ψ‖X,·∧σ M ∞ is adapted and .P − a.s. continuous, as .

| | |‖ | ‖ n ‖ ‖ | | |‖ | nj ‖ j‖ Ψ − Ψ ≤ sup sup |‖Ψ‖X,r∧σ∞ M − ‖Ψ | | M M X,r∧σ X,r∧σ | ∞

r∈[0,T ]

∞

r∈[0,T ]

‖ ‖ = ‖Ψ − Ψ nj ‖X,σ M , ∞

which has .P − a.s. limit as .j → ∞ equal to zero. Thus .‖Ψ‖X,·∧σ∞ M is given, P−a.s., as the uniform in time limit of adapted and continuous processes, verifying the result. Moving on, it is now that we make use of (4.15) much in the same way as M and .δ := σ M − σ M . we did for (4.14). This will be done in the context of .γ := σ∞ j ∞ j Indeed for any .j ∈ N we can choose an .mj ∈ N (where .mj = nl some l) such that for all .k ≥ mj ,

.

⎞ ⎛‖ ‖ ‖ ‖2 2 sup E ‖Ψ n ‖X,σ M ∧τ M,T − ‖Ψ n ‖X,σ M ∧τ M,T ≤ 2−2j .

.

k

n∈N

∞

n

n

In particular, through a relabeling of .σmMj = σlM , ⎞ ⎛ ‖ ‖2 ‖2 ‖ E ‖Ψ mj ‖X,σ M − ‖Ψ mj ‖X,σ M ≤ 2−2j

.

∞

mj

M ≤ σ M ≤ σ M ≤ τ M,T . In a familiar way by choosing n as .mj and using that .σ∞ mj mj k we define ⎫ ⎧ ‖ m ‖2 ‖ m ‖2 ' −(j +2) j‖ ‖ Ψ .Ωj := ‖Ψ j ‖ − < 2 X,σ M X,σ M ∞

mj

so that, just as we showed for (4.20), ˇ K := Ω

∞ ⋂

.

Ω'j ,

j =K

ˇ := Ω

∞ ⋃

ˇ K, Ω

⎛ ⎞ ˇ = 1. P Ω

K=1

ˆ ∩ Ω, ˇ For arbitrary given .R > 0, we find a constant M such that at every .ω ∈ Ω 2 M R,T M either .σ∞ = T or .‖Ψ‖X,σ M ≥ R. In both instances it is clear that .τ ≤ σ∞ , ∞

4.3 Cauchy Criteria

103

M < T. ˆ ∩Ω ˇ such that .σ∞ thus proving the proposition. To this end we fix an .ω ∈ Ω M is the decreasing limit of .(σ M ), then for sufficiently large .m we must also As .σ∞ j mj have that .σmMj < T . Exactly as in (4.22),

‖ ‖ m ‖ ‖ ‖Ψ j ‖ M = (M˜ − 1 + 2−j ) + ‖Ψ mj ‖ . X,σ X,0

.

mj

(4.26)

From the proven convergence we also have that for sufficiently large .mj , ‖ ‖ ‖Ψ − Ψ mj ‖ M < 1, X,σ

.

∞

(4.27)

m ˇ K for some which implies that .‖Ψ‖X,σ∞ M > ‖Ψ j ‖X,σ M − 1, and likewise as .ω ∈ Ω ∞ K,

‖ ‖ m ‖2 ‖ ‖Ψ j ‖ M − ‖Ψ mj ‖2 M < 1. X,σ X,σ

.

mj

∞

(4.28)

We fix an .mj large enough so that (4.26), (4.27), and (4.28) all hold. Substituting (4.26) into (4.28) gives that ⎛ ‖ ‖ m ‖2 ‖ ⎞2 ‖Ψ j ‖ M > (M˜ − 1 + 2−j ) + ‖Ψ mj ‖ − 1 > (M˜ − 1)2 − 1. X,σ X,0

.

∞

If .M˜ > 2, then the expression on the right is positive and ⎛ ⎞1 ‖ m ‖ ‖Ψ j ‖ M > (M˜ − 1)2 − 1 2 . X,σ

.

∞

Furthermore, ⎛ ⎞1 ˜ − 1)2 − 1 2 − 1, ‖Ψ‖X,σ∞ M > (M

.

where the right hand side is of course monotone increasing and unbounded in .M˜ and hence M. By choosing M large enough such that ⎡⎛ .

we complete the proof.

⏋2 ⎞1 2 (M˜ − 1)2 − 1 − 1 > R, ⨆ ⨅

104

4 A Toolbox for Nonlinear SPDEs

4.4 Enhanced Regularity and an Energy Equality Having established a framework for SPDEs in Chap. 3, we introduce techniques to facilitate our analysis in it. The Itô Formula is well regarded as one of the most useful tools in stochastic analysis, and we establish what can be considered as a specific case of this result in our framework. We shall introduce a new setting in which the established solution framework falls, with the understanding that we would like to apply this to solutions while also using the results to deduce the existence of solutions when they are not a priori known. This is well understood in the typical variational framework, for which we again refer to [51, 57], but we take care in addressing subtle changes. The first is the loss of the duality structure, though for this result we do assume a bilinear form relation which behaves similarly. The second is that we conduct the proof for local solutions, necessary for our motivating class of equations, so it is important for us to explicitly address how the localization affects the proof. Indeed, the consideration of local solutions, as well as the related localization in the construction of the integral, martingale theory, and analytical techniques, is an important extension of the framework of [51, 56, 57]. Similarly the final key change is that we do not assume any integrability over the probability space of our processes, demanding again another source of localization which we find worthy of detailing. To prove this energy equality we shall rely on looking at partitions in time over which some nice properties are satisfied, before taking the limit as the increments go to zero. Toward this, we recall the following lemma from [57, Lemma 4.2.6]. This section follows the ideas of [57, Lemma 4.2.5]. Lemma 4.4 Let .X1 , X2 be two Banach spaces with continuous embedding .X1 ͨ− → X2 and suppose that for some .T > 0 and a stopping time .τ , .Ф : Ω × [0, T ] → X2 is such that for .P − a.e. .ω, .Ф· (ω) ∈ C ([0, T ]; X2 ) and .Ф· 1·≤τ ∈ L2 (Ω × [0, T ]; X1 ). Then for any .A ⊂ (0, T ) with .λ(A) = 0 there exists a sequence of partitions .(Il ) such that: { } 1. .Il := 0 = t0l < t1l < · · · < tkl l = T , .maxj |tjl − tjl −1 | → 0 as .l → ∞. 2. .Il ⊂ Il+1 . 3. .Il ∩ A = ∅. 4. For .P − a.e. .ω and every .tjl with .1 ≤ j ≤ kl−1 , .Фt l (ω)1t l ≤τ (ω) ∈ X1 . j

l

j

l

˜ defined at each .t ∈ [0, T ] and .ω ∈ Ω by ˆ ,Ф 5. The processes .Ф ˆ lt (ω) := .Ф

kl ⎲ j =2

˜ lt (ω) := Ф

kl−1 ⎲ j =1

1[t l

(t)Фt l (ω)1t l

1[t l

(t)Фt l (ω)1t l ≤τ (ω)

l j −1 ,tj )

l j −1 ,tj )

j −1 ≤τ (ω)

j −1

j

j

,

4.4 Enhanced Regularity and an Energy Equality

105

belong to .L2 (Ω × [0, T ]; X1 ) and both converge to .Ф· 1·≤τ in this space. Before moving on, we take a moment to discuss this result. In the statement of [57, Lemma 4.2.6], there is no continuity assumption on .Ф, and indeed, this is surplus to requirement; however we want to make explicit that it is genuinely the process .Ф taken in item 5 and not some other representative of an equivalence class. The assumptions are of course reminiscent of Definition 3.4 and just as was stressed there that the progressively measurable process in V was not necessarily the continuous process itself but just a .P × λ − a.s. equivalent representation, we are reminded again that elements of .L2 (Ω × [0, T ]; X1 ) are only an equivalence class ˜ l are significant ˆ l, Ф of .P × λ − a.s. equal functions. Therefore the representations .Ф as they are piecewise constant in the space .X1 . We impose some additional structure on the established framework to conduct the analysis of this section. We work with a triple of embedded Hilbert spaces V ͨ− → H ͨ− → U,

.

where the embeddings are continuous, V is assumed dense in H , and there exists a continuous bilinear form .〈·, ·〉U ×V : U ×V → R such that for every .φ ∈ H, ψ ∈ V , 〈φ, ψ〉U ×V = 〈φ, ψ〉H .

.

We suppose that for some .T > 0 and stopping time .τ : 1. 2. 3. 4. 5.

Ψ 0 ∈ L2 (Ω; H ) is .F0 -measurable. 2 .η ∈ L (Ω × [0, T ]; U )). H .B ∈ I (W). 2 .Ψ · 1·≤τ ∈ L (Ω × [0, T ]; V )) and is progressively measurable in V . The identity .

⎰

t∧τ

Ψt = Ψ0 +

.

0

⎰ ηs ds +

t∧τ

Bs dWs

(4.29)

0

holds .P − a.s. in U for all .t ∈ [0, T ]. Remark 4.2 Once more, it is clear from assumption 5 that for .P − a.e. .ω, .Ψ(ω) ∈ C ([0, T ]; U ), but the progressively measurable process assumed in item 4 is only equivalent to this .Ψ · 1·≤τ .P × λ − a.s.. It is a slight abuse of notation commonplace in the literature, and we shall follow suit without excessive clarification at each use. With this structure in place, we first look to deduce some improved regularity on Ψ. We are motivated by applications to the existence theory of solutions, and if .η, B were given by functions of .Ψ, then note that .Ψ is nearly a local strong solution of the corresponding SPDE. The missing ingredient is pathwise continuity in H , which we look to deduce from only the given properties.

.

106

4 A Toolbox for Nonlinear SPDEs

For this we fix an application of Lemma 4.4 relative to the assumptions laid out above. We take T and .τ as in the assumptions,⎛ .X1 = V , .X⎞2 = U , .Ф = Ψ. From ‖ ‖2 item 4 we have that for .λ − a.e. .t ∈ [0, T ], .E ‖Ψ t 1t≤τ ‖V < ∞, and we take A l

l

˜ , I l are defined as in ˆ ,Ψ to be the .λ-zero set on which this does not hold. Then .Ψ Lemma 4.4, and we define ⋃ .I := I l. l∈N

Lemma 4.5 We have that ⎛ E

⎞ sup

.

t∈[0,T ]

‖Ψ t ‖2H

< ∞.

Proof For .P − a.e. .ω and every .s < t with .s, t ∈ I ∩ [0, τ ]/{T }, observe that ‖⎰ t ‖2 ‖2 ‖ \ / ⎰ t ⎰ t ‖ ‖ ‖ ‖ ‖ Br dWr ‖ − ‖Ψ t − Ψ s − ‖ B dW + 2 Ψ , B dW r r‖ s r r ‖ ‖ ‖

.

s

s

H

s

H

‖2 ‖2 ‖⎰ t ‖⎰ t ‖ ‖ ‖ ‖ 2 ‖ ‖ ‖ = ‖ Br dWr ‖ − ‖Ψ t − Ψ s ‖H − ‖ Br dWr ‖ ‖ s s H H \ \ / / ⎰ t ⎰ t + 2 Ψt − Ψs , Br dWr + 2 Ψ s , Br dWr H

s

s

\ / ⎰ t = 2 Ψt , Br dWr − ‖Ψ t − Ψ s ‖2H

H

H

H

s

/ \ ⎰ t = 2 Ψt , Ψt − Ψs − ηr dr − ‖Ψ t ‖2H − ‖Ψ s ‖2H + 2〈Ψ t , Ψ s 〉H s

.

H

/ \ ⎰ t = ‖Ψ t ‖2H − ‖Ψ s ‖2H − 2 Ψ t , ηr dr s

⎰ = ‖Ψ t ‖2H − ‖Ψ s ‖2H − 2

t s

H

〈ηr , Ψ t 〉U ×V dr,

which we rewrite as the equality ⎰ ‖Ψ t ‖2H − ‖Ψ s ‖2H = 2

t

.

s

\ / ⎰ t 〈ηr , Ψ t 〉U ×V dr + 2 Ψ s , Br dWr

H

s

‖2 ‖2 ‖⎰ t ‖ ⎰ t ‖ ‖ ‖ ‖ ‖ ‖ ‖ + ‖ Br dWr ‖ − ‖Ψ t − Ψ s − Br dWr ‖ ‖ . s

H

s

H

4.4 Enhanced Regularity and an Energy Equality

107

Note that we have had to exclude T from this set, as one does not know if .Ψ T ∈ V P − a.s.. Using this equality, for any .l ∈ N and .t = til ∈ Il ∩ (0, τ ]/{T },

.

2 .‖Ψ t ‖H

− ‖Ψ 0 ‖2H

i−1 ⎛‖ ‖ ‖2 ⎞ ‖2 ⎲ ‖ ‖ ‖ ‖ = ‖Ψ t l ‖ − ‖Ψ t l ‖ j +1

j =0 i−1 ⎛ ⎰ ⎲ 2 = j =0

=2

/

/

\ ηr , Ψ t l

j +1

tjl

H

U ×V

⎰

dr + 2 Ψ t l , j

\

tjl +1

Br dWr

tjl

H

‖2 ‖2 ⎞ ‖⎰ l ‖ ⎰ tl ‖ ‖ ‖ tj +1 ‖ j +1 ‖ ‖ ‖ ‖ +‖ Br dWr ‖ − ‖Ψ t l − Ψ t l − Br dWr ‖ j ‖ ‖ ‖ tl ‖ j +1 tl

⎛ i−1 ⎰ ⎲

j

tjl +1

j

H

/

⎰

\ ηr , Ψ t l

j +1

tjl

j =0

=2

tjl +1

j

H

U ×V

dr +

tjl +1

/

\ Br , Ψ t l

j

tjl

H

⎞

H

dWr

⎛‖ ‖2 ‖2 ⎞ ‖ ⎰ tl i−1 ‖⎰ t l ‖ ‖ ‖ ⎲ j +1 j +1 ‖ ‖ ‖ ‖ ⎝‖ Br dWr ‖ − ‖Ψ t l − Ψ t l − Br dWr ‖ ⎠ + j l ‖ tl ‖ ‖ ‖ j +1 t

⎛ i−1 ⎰ ⎲

j =0 tjl +1

j

/

˜ lr ηr , Ψ

tjl

j =0

⎰

t1l

+2 0

j

H

⎞

\ U ×V

dr + 2

⎛ i−1 ⎰ ⎲ j =1

tjl +1

tjl

/

ˆ lr Br , Ψ

H

⎞

\ H

dWr

〈Br , Ψ 0 〉H dWr

⎛‖ ‖2 ‖2 ⎞ ‖ ⎰ tl i−1 ‖⎰ t l ‖ ‖ ‖ ⎲ j +1 j +1 ‖ ‖ ‖ ‖ ⎝‖ + Br dWr ‖ − ‖Ψ t l − Ψ t l − Br dWr ‖ ⎠ j l ‖ tl ‖ ‖ ‖ j +1 t j =0

=2

⎰ t/ 0

˜ lr ηr , Ψ

j

\ U ×V

j

H

dr + 2

⎰ t/ 0

ˆ lr Br , Ψ

\ H

⎰

t1l

dWr + 2 0

H

〈Br , Ψ 0 〉H dWr

⎛‖ ‖2 ‖2 ⎞ ‖ ⎰ tl i−1 ‖⎰ t l ‖ ‖ ‖ ⎲ j +1 j +1 ‖ ‖ ‖ ⎝‖ + Br dWr ‖ − ‖Ψ t l − Ψ t l − Br dWr ‖ ⎠ , ‖ j ‖ tl ‖ ‖ ‖ j +1 tl j =0

j

j

H

H

(4.30) where we have applied Proposition 2.17 and Remark 2.4. In particular we have that ‖Ψ t ‖2H ≤ ‖Ψ 0 ‖2H + 2

⎰ t/

.

0

l

˜r ηr , Ψ

\ U ×V

dr + 2

⎰ t/ 0

l

ˆr Br , Ψ

\ H

dWr

108

4 A Toolbox for Nonlinear SPDEs

⎰

t1l

+2 0

⎛‖ ‖2 ⎞ i−1 ‖⎰ t l ‖ ⎲ j +1 ‖ ⎝‖ 〈Br , Ψ 0 〉H dWr + Br dWr ‖ ⎠ . ‖ ‖ tl ‖ j =0

j

H

Our goal is to show that ⎞

⎛ E

max

.

t∈Il ∩(0,τ ]/{T }

‖Ψ t ‖2H

≤c

for some constant c independent of l. To this end, observe that ⎞

⎛ E

.

max

t∈Il ∩(0,τ ]/{T }

‖Ψ t ‖2H

|/ | ⎞ \ | | | ηr , Ψ | dr ˜ lr | U ×V | 0 |⎞ / \ | l ˆ Br , Ψ r dWr || H

⎛⎰ ⎛ ⎞ ≤ E ‖Ψ 0 ‖2H + 2E ⎛ + 2E ⎛ + 2E ⎡

|⎰ | sup ||

t∈[0,T ]

0

|⎰ | sup ||

t∈[0,T ]

t∧τ

t∧τ

0

T ∧τ

|⎞ | 〈Br , Ψ 0 〉H dWr ||

⎛‖ ‖2 ⎞⎤ i−1 ‖⎰ t l ‖ ⎲ j +1 ‖ ⎝‖ + E⎣ max Br dWr ‖ ⎠⎦ . ‖ ‖ tl ‖ t∈Il ∩(0,τ ]/{T } j =0

j

H

We shall treat each term individually. First, we have that ⎛⎰

T ∧τ

2E

.

0

|/ | ⎞ \ | | | ηr , Ψ | dr ˜ lr | U ×V | ⎛⎰ T ∧τ ‖ ‖2 ⎞ ‖˜l‖ 2 ‖ηr ‖U + ‖Ψ ≤E r ‖ dr V

0

⎛⎰

T ∧τ

≤E 0

⎞ ‖ηr ‖2U dr

⎛⎰

T ∧τ

+ max E m≤L

0

‖ ‖2 ⎞ ‖ ˜ m‖ ‖Ψ r ‖ dr + 1, V

where L is taken sufficiently large so that for all .m ≥ L, ⎛⎰

|‖ ‖ | ⎞ |‖ m ‖2 | |‖Ψ ˜ r ‖ − ‖Ψ r ‖2V | dr ≤ 1 . | | V 2

T ∧τ

E

.

0

For the first stochastic integral we apply the classical Burkholder–Davis–Gundy Inequality, Theorem A.5, seeing that ⎛ 2E

.

|⎰ | sup ||

t∈[0,T ]

t∧τ 0

/

ˆ lr Br , Ψ

|⎞ | dWr || H

\

4.4 Enhanced Regularity and an Energy Equality

⎛⎰ ≤ cE

T ∧τ

≤ cE

T ∧τ

‖ l ‖2 ⎞ 12 ‖ˆ ‖ ‖Br ‖L 2 (U;H ) ‖Ψ r ‖ dr H

0

sup

r∈[0,T ∧τ ]

⎛ = cE

sup

r∈[0,T ∧τ ]

1 = E 2 =

1 E 2

‖ l ‖2 ⎰ ‖ˆ ‖ ‖Ψ r ‖ H

‖ l ‖2 ‖ˆ ‖ ‖Ψ r ‖

⎛

sup

⎛

dr

2

⎛ ≤ cE

⎞ 12

‖/ \ ‖2 ‖ ˆ lr ‖ ‖ Br , Ψ ‖

H L 2 (U;R)

0

⎛⎰

109

r∈[0,T ∧τ ]

T ∧τ 0

2

‖Br ‖2L 2 (U;H ) dr

⎞ 1 ⎛⎰ 2

H

‖ l ‖2 ‖ˆ ‖ ‖Ψ r ‖

⎞1

T ∧τ 0

⎞

⎛⎰ 0

⎞

t∈Il ∩(0,τ ]/{T }

T ∧τ

+ cE

H

sup

‖Br ‖L 2 (U;H ) dr 2

‖Ψ t ‖2H

⎛⎰ + cE

⎞ 12 ⎞

‖Br ‖2L 2 (U;H ) dr

T ∧τ

0

⎞ ‖Br ‖2L 2 (U;H ) dr ,

where c here is a generic constant changing from line to line, independent of l. Putting this together, we now see that 1 . E 2

⎞

⎛ max

t∈Il ∩(0,τ ]/{T }

⎛

≤E

‖Ψ t ‖2H

‖Ψ 0 ‖2H ⎛⎰

+ max E m≤L

⎛

+ 2E ⎡

⎞

⎛⎰

⎞

T ∧τ

+E

‖ηr ‖2U dr

0

T ∧τ

⎛⎰ ‖ ‖2 ⎞ ‖ ˜ m‖ ‖Ψ r ‖ dr + 1 + cE V

0

|⎰ t |⎞ | | sup || 〈Br , Ψ 0 〉H dWr ||

t∈[0,T ]

T ∧τ 0

⎞ ‖Br ‖L 2 (U;H ) dr 2

0

⎛‖ ‖2 ⎞⎤ i−1 ‖⎰ t l ‖ ⎲ j +1 ‖ ⎝‖ + E⎣ max Br dWr ‖ ⎠⎦ . ‖ ‖ tl ‖ t∈Il ∩(0,τ ]/{T } j =0

j

H

For the stochastic integral involving the initial condition, we can treat this identically to generate the bound ⎛ 2E

.

|⎰ | sup ||

t∈[0,T ]

0

|⎞ ⎛⎰ ⎛ ⎞ | 〈Br , Ψ 0 〉H dWr || ≤ E ‖Ψ 0 ‖2H +cE

⎞ ‖Br ‖2L 2 (U;H ) dr .

T ∧τ

t∧τ

0

110

4 A Toolbox for Nonlinear SPDEs

As for the final term, it is clear that the supremum over all partitions can be bounded by taking the partition for .t = tkl l = T . Thus ⎡

⎛‖ ⎡ ⎛‖ ‖2 ⎞⎤ ‖2 ⎞⎤ k⎲ i−1 ‖⎰ t l l −1 ‖⎰ tjl +1 ‖ ‖ ⎲ j +1 ‖ ‖ ⎝‖ ⎝‖ .E ⎣ max Br dWr ‖ ⎠⎦ ≤ E ⎣ Br dWr ‖ ⎠⎦ ‖ ‖ ‖ tl ‖ tl ‖ ‖ t∈Il ∩(0,τ ]/{T } j =0

j

j =0

H

j

H

⎛‖ ‖2 ⎞ k⎲ l −1 ‖⎰ tjl +1 ‖ ‖ ‖ = E ⎝‖ Br dWr ‖ ⎠ ‖ tl ‖ j =0

=

j

k⎲ l −1

⎛⎰ E

=E

tjl +1

tjl

j =0

⎛⎰

H

T

0

⎞ ‖Br ‖L 2 (U;H ) dr 2

⎞ ‖Br ‖L 2 (U;H ) dr 2

having applied Proposition 2.15. In total then we have that ⎞

⎛ E

.

max

t∈Il ∩(0,τ ]/{T }

‖Ψ t ‖2H

≤ 4E

⎛

‖Ψ 0 ‖2H

⎞

⎛⎰

+ 2E

T

+ cE 0

‖ηr ‖2U dr

‖ ‖2 ⎞ ‖ ˜ m‖ ‖Ψ r ‖ dr + 2 V

0

⎛⎰

⎞

T ∧τ

0

T ∧τ

+ 2 max E m≤L

⎛⎰

⎞

‖Br ‖L 2 (U;H ) dr , 2

which gives a finite bound independent of l. As .Il ⊂ Il+1 , then the sequence .

max

t∈Il ∩(0,τ ]/{T }

‖Ψ t ‖2H

is .P − a.s. monotone increasing in l, so we can apply the Monotone Convergence Theorem to see that ⎛ ⎞ ⎞ ⎛ E

.

sup

t∈I ∩(0,τ ]/{T }

‖Ψ t ‖2H

= lim E l→∞

max

t∈Il ∩(0,τ ]/{T }

‖Ψ t ‖2H

< ∞.

(4.31)

Thus for .P − a.e. .ω, we have that .

sup

t∈I ∩(0,τ (ω)]/{T }

‖Ψ t (ω)‖2H = c˜ < ∞

and .Ψ · (ω) ∈ C([0, T ]; U ). We fix such an .ω and any .t ∈ [0, τ (ω)]. As the mesh of the partitions goes to zero, then there is a sequence of times .(tn ) in .I ∩(0, τ (ω)]/{T }

4.4 Enhanced Regularity and an Energy Equality

111

( ) such that .tn −→ t. The sequence . Ψ tn (ω) is uniformly bounded in H so admits a weakly convergent subsequence in this space, to a limit which we call .ψ. From the continuous embedding of H into U , this weak (convergence also holds in U , but ) from the continuity of .Ψ · (ω) in U , we have that . Ψ tn (ω) converges strongly and therefore weakly to .Ψ t (ω) in U . By the uniqueness of limits in the weak topology, we conclude that .Ψ t (ω) = ψ and thus belongs to H . Moreover, the weak limit preserves the boundedness in H , so .‖Ψ t (ω)‖H ≤ c. ˜ Therefore sup

.

t∈I ∩(0,τ (ω)]/{T }

‖Ψ t (ω)‖2H =

sup t∈[0,τ (ω)]

‖Ψ t (ω)‖2H

for our fixed .ω in a full measure set, and thus .P−a.s.. We also know that .Ψ · = Ψ ·∧τ from the identity (4.29), so this equality extends to ‖Ψ t (ω)‖2H = sup ‖Ψ t (ω)‖2H .

sup

.

t∈[0,T ]

t∈I ∩(0,τ (ω)]/{T }

⨆ ⨅

Combining this with (4.31) concludes the proof.

Having now justified that for .P − a.e. .ω and all .t ∈ [0, T ] that .Ψ t (ω) ∈ H , we move on to prove weak continuity in this space. Lemma 4.6 For .P − a.e. .ω, .Ψ · (ω) is weakly continuous in H . Proof We fix .t ∈ [0, T ] but now take any sequence of times .(tn ) such that .tn → t. For .P − a.e. .ω and any given .φ ∈ H we must justify that .

〈 〉 lim Ψ tn (ω) − Ψ t (ω), φ H = 0.

(4.32)

n→∞

For any .ε > 0, from the density of V in H , there exists .φ k ∈ V such that ‖ ‖ ‖ ‖ ‖φ − φ k ‖

.

H


ε t∈[0,T ]

≤

1 E (MT ) . ε

Proof See [44, Theorem 1.3.8].

⨆ ⨅

Theorem A.1 (Doob–Meyer Decomposition) Let .M ∈ M2c . Then there exists a unique (up to indistinguishability) continuous, adapted, nondecreasing real valued process .[M] with .[M]0 = 0 (.P − a.s.) such that © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 D. Goodair, D. Crisan, Stochastic Calculus in Infinite Dimensions and SPDEs, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-69586-5

127

128

Appendix A

M 2 − [M]

.

is a real valued martingale. ⨆ ⨅

Proof See [44, Theorem 1.4.10].

Theorem A.2 Let .M, N ∈ Mc . Then there exists a unique (up to indistinguishability) continuous, adapted, bounded-variation process .[M, N ] with .[M, N]0 = 0 (.P − a.s.) such that MN − [M, N ]

.

is a real valued martingale. ⨆ ⨅

Proof See [44, Theorem 1.5.13]. M2c

Lemma A.1 Let .M ∈

be of bounded-variation. Then M is constant .P − a.s.

Proof This constitutes part of the proof of [44, Theorem 1.5.13]. Lemma A.2 Let .M, N ∈

M2c .

⨆ ⨅

Then .P − a.s. for any .t ≥ 0,

[M, N]2t ≤ [M]t [N]t .

.

⨆ ⨅

Proof See [44, Problem 1.5.7]. Lemma A.3 Let .M, N ∈

M2c .

Then .P − a.s. for any .T ≥ 0,

VRT ([M, N]) ≤

.

1 ([M]t + [N]t ) . 2

Proof See [44, Problem 1.5.7]. Theorem A.3 Let .M ∈ M2c , and consider any sequence of partitions { } Il := 0 = t0l < t1l < · · · < tkl l = T

.

with .maxj |tjl − tjl −1 | → 0 as .l → ∞. Then for all .t ∈ [0, T ], for any .ε > 0, ⎫⎞ | ⎛ ⎧| | | ⎪ ⎪ ⎬ | |2 ⎨| ⎲ | ⎟ | | | ⎜ | . lim P ⎝ | |Mt l − Mt l | − [M]t | > ε ⎠ = 0. j j +1 ⎪ | | ⎪ l→∞ ⎭ ⎩|t l ≤t | j +1 If, in addition, .|M|, [M] ∈ L∞ (Ω × [0, T ]; R), then

⨆ ⨅

Appendix A

129

|⎞ ⎛| | | | |2 |⎲ | |⎟ | | ⎜| . lim E ⎝| |Mt l − Mt l | − [M]t |⎠ = 0. j j +1 | |l l→∞ | |tj +1 ≤t ⨆ ⨅

Proof See [44, Theorem 1.5.8].

Theorem A.4 (Lévy’s Characterization of Brownian Motion) Let M be a real valued continuous local martingale with .M0 = 0 .P − a.s., and .[M]t = t. Then M is a Brownian Motion. ⨆ ⨅

Proof See [44, Theorem 1.3.16].

Theorem A.5 (Burkholder–Davis–Gundy Inequality) For every .p ≥ 1, there exists a constant .Cp such that, for every real valued continuous local martingale M with .M0 = 0 .P − a.s., and for any stopping time .τ ≥ 0, ⎛ E

.

⎞ sup |Mt |

t∈[0,τ ]

p

⎞ ⎛ p ≤ Cp E [M]τ2 .

Proof See [10].

⨆ ⨅

A.2 Classical Tightness Criteria Theorem A.6 Let (ψ n ) be a sequence of processes in D ([0, T ]; R). Suppose that for any sequence of stopping times (γn ), γn : Ω → [0, T ], and constants (δn ), δn ≥ 0, and δn → 0 as n → ∞: 1. For every t ∈ [0, T ], the sequence of the laws of (ψ nt ) is tight in the space of probability measures over R.⎛{ | | }⎞ | | 2. For every ε > 0, limn→∞ P ω ∈ Ω : |ψ n(γn +δn )∧T − ψ nγn | > ε = 0. Then the sequence of the laws of (ψ n ) is tight in the space of probability measures over D ([0, T ]; R). Proof See [3, Theorem 1].

⨆ ⨅

Theorem A.7 Let E be a metric space and F a collection of functions in C (E; R) with the property that F separates points in E and is closed under addition. Let (μn ) be a sequence of probability measures on D ([0, T ]; E) satisfying the following: 1. For each ε > 0, there exists a compact K ⊂ E such that for every n ∈ N, μn (D ([0, T ]; K)) > 1 − ε. 2. For every f ∈ F, defining the mapping f˜ : D ([0, T ]; E) → D ([0, T ]; R) by [f˜(φ)](t) = f [φ(t)], then the sequence (μn ◦ f˜−1 ) is tight in the space of probability measures over D ([0, T ]; R).

130

Appendix A

Then the sequence (μn ) is tight in the space of probability measures over D ([0, T ]; E). ⨆ ⨅

Proof See [43, Theorem 3.1].

A.3 Stochastic Grönwall Lemma Continuing to look at techniques from PDE theory, a Stochastic Grönwall Lemma will prove of great significance in applications. While in some situations we can apply the classical Grönwall Lemma to the expectation of the process, this is complicated when we have control by the expectation of a product of processes. To overcome this Glatt-Holtz and Ziane proved the following: Lemma A.4 Fix .t > 0 and suppose that .φ, ψ, η are real valued, nonnegative stochastic processes. Assume, moreover, that there exist constants .c' , cˆ (allowed to depend on t) such that for .P − a.e. .ω, ⎰

t

.

0

ηs (ω)ds ≤ c' ,

(A.1)

and for all stopping times .0 ≤ θj < θk ≤ t, ⎛ E

.

⎞ sup φ r

r∈[θj ,θk ]

⎛⎰ +E

⎞

θk

θj

⎛

⎛ ⎞ ⎰ φ θj + 1 + ≤ cE ˆ

ψ s ds

⎞

θk

θj

ηs φ s ds

< ∞.

Then there exists a constant C dependent only on .c' , c, ˆ t such that ⎛ E

.

⎞ sup φ r

r∈[0,t]

Proof See [32, Lemma 5.3].

⎛⎰ +E 0

⎞

t

ψ s ds

⎾ ⏋ ≤ C E(φ 0 ) + 1 . ⨆ ⨅

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Index

C Cauchy convergence, 97–103 Constant multiplicative noise, 123–126 Cross-variation characterisation, 38 Cross-variation convergence, 43 Cross-variation in finite dimensions, 43 Cross-variation in infinite dimensions, 43 Cross-variation of local martingales, 1, 38–45 Cylindrical BM, 1, 7, 23–27, 45–47, 49, 53, 55, 57, 59, 80, 83

D Dual representation for Itô Integral w.r.t BM, 8, 10, 13–15, 17, 74, 123, 124, 126 Dual representation for Itô Integral w.r.t Cylindrical BM, 45–56 Dual representation for local Itô Integral w.r.t BM, 8, 10, 13–15, 17, 74, 123, 124, 126

E Energy equality for solutions, 105, 107, 109, 111, 113, 115, 117, 119, 121 Enhanced regularity of solutions, 105, 107, 109, 111, 113, 115, 117, 119, 121 Existence for Finite Dimensional SPDEs, 83–92 Existence of maximal solutions, 63–70

G Galerkin Approximation, 83, 96, 97

I Indistinguishability, 31, 64, 127 Itô Integration of simple processes, 7–9, 11, 12, 16, 18, 19, 30, 51 Itô Integration w.r.t BM, 8, 10, 13–15, 17, 74, 123, 124, 126 Itô Integration w.r.t Cylindrical BM, 45–56 Itô Integration w.r.t local martingale, 16 Itô Isometry for simple processes, 11 Itô Isometry w.r.t BM, 10, 13, 14 Itô Isometry w.r.t Cylindrical BM, 45–56 Itô Isometry w.r.t multi-dimensional BM, 13 Itô local strong solutions, 2, 59–63 Itô-Stratonovich conversion, 4, 44, 60, 78, 80 Itô strong solutions, 2, 59–63 Itô weak solutions, 77–80

L Levy’s Characterisation of BM, 124, 125, 129

M Martingale convergence, 32, 43, 53, 54, 97 Martingale property of Itô Integral w.r.t BM, 14, 17 Martingale property of Itô Integral w.r.t Cylindrical BM, 47, 48, 51–54 Martingale property of Itô Integral w.r.t local martingale, 15–23, 30, 45, 48, 62 Martingales in Hilbert Spaces, 27–45 Martingale weak solutions, 77–80 Maximal solutions, 63–70

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 D. Goodair, D. Crisan, Stochastic Calculus in Infinite Dimensions and SPDEs, SpringerBriefs in Mathematics, https://doi.org/10.1007/978-3-031-69586-5

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136

Index

P Progressive measurability, 8, 11, 14–19, 21, 22, 46, 47, 49, 50, 61–67, 72, 76, 78, 80, 81, 84, 105

Stratonovich local strong solutions, 76 Stratonovich strong solutions, 59–66, 70–72, 76, 78, 80 Stratonovich weak solutions, 77–80

Q Q-Cylindrical BM, 23–27, 45–46, 57, 59, 80, 83 Quadratic variation convergence, 15, 31, 36, 38, 40, 45, 52, 56, 85, 124 Quadratic variation in Hilbert Spaces, 38 Quadratic variation of Itô Integral w.r.t BM, 124 Quadratic variation of Itô Integral w.r.t Cylindrical BM, 52, 56 Quadratic variation representation, 36

T Tightness in .L2 space, 92, 95 Tightness in Skorohod space, 96

R Regular Q-Cylindrical BM, 23–27

S Simple processes, 7–9, 11, 18, 19 Stochastic Dominated Convergence w.r.t BM, 20 Stochastic Dominated Convergence w.r.t Cylindrical BM, 51 Stochastic Grönwall Lemma, 130 Stratonovich Integration w.r.t Cylindrical BM, 57 Stratonovich Integration w.r.t local martingale, 57, 71, 72, 76

U Uniqueness for Finite Dimensional SPDEs, 83–92 Unique solutions, 63, 64, 70

V Variational framework, 2, 3, 104 2-variation of Itô Integral w.r.t BM, 8, 10, 13–15, 17, 74, 123, 124, 126 2-variation of Itô Integral w.r.t Cylindrical BM, 45–56

W Weak and norm continuity, 28, 120 Weak continuity of solutions, 30, 111, 116 Weak equicontinuity, 97 Weak-strong solutions, 59–63, 77–80

Z Zorn’s Lemma, 66, 70