Bichteler Klaus - Stochastic Integration with Jumps.pdf - Matematyka - Asfoora

Contents

Preface ............................................................ xi

Chapter 1 Introduction ............................................. 1

1.1 Motivation: Stochastic Dierential Equations ............... 1

The Obstacle 4, Ito’s Way Out of the Quandary 5, Summary: The Task Ahead 6

1.2 Wiener Process ............................................. 9

Existence of Wiener Process 11, Uniqueness of Wiener Measure 14, Non-

Dierentiability of the Wiener Path 17, Supplements and Additional Exercises 18

1.3 The General Model ........................................ 20

Filtrations on Measurable Spaces 21, The Base Space 22, Processes 23, Stop-

ping Times and Stochastic Intervals 27, Some Examples of Stopping Times 29,

Probabilities 32, The Sizes of Random Variables 33, Two Notions of Equality for

Processes 34, The Natural Conditions 36

Chapter 2 Integrators and Martingales ............................. 43

Step Functions and Lebesgue–Stieltjes Integrators on the Line 43

2.1 The Elementary Stochastic Integral ........................ 46

Elementary Stochastic Integrands 46, The Elementary Stochastic Integral 47, The

Elementary Integral and Stopping Times 47, L p -Integrators 49, Local Properties 51

2.2 The Semivariations ........................................ 53

The Size of an Integrator 54, Vectors of Integrators 56, The Natural Conditions 56

2.3 Path Regularity of Integrators ............................. 58

Right-Continuity and Left Limits 58, Boundedness of the Paths 61, Redenition of

Integrators 62, The Maximal Inequality 63, Law and Canonical Representation 64

2.4 Processes of Finite Variation ............................... 67

Decomposition into Continuous and Jump Parts 69, The Change-of-Variable

Formula 70

2.5 Martingales ............................................... 71

Submartingales and Supermartingales 73, Regularity of the Paths: Right-

Continuity and Left Limits 74, Boundedness of the Paths 76, Doob’s Optional

Stopping Theorem 77, Martingales Are Integrators 78, Martingales in L

p 80

Chapter 3 Extension of the Integral ................................ 87

Daniell’s Extension Procedure on the Line 87

3.1 The Daniell Mean ......................................... 88

A Temporary Assumption 89, Properties of the Daniell Mean 90

3.2 The Integration Theory of a Mean ......................... 94

Negligible Functions and Sets 95, Processes Finite for the Mean and Dened Almost

Everywhere 97, Integrable Processes and the Stochastic Integral 99, Permanence

Properties of Integrable Functions 101, Permanence Under Algebraic and Order

Operations 101, Permanence Under Pointwise Limits of Sequences 102, Integrable

Sets 104

vii

Contents

viii

3.3 Countable Additivity in p -Mean .......................... 106

The Integration Theory of Vectors of Integrators 109

3.4 Measurability ............................................ 110

Permanence Under Limits of Sequences 111, Permanence Under Algebraic and

Order Operations 112, The Integrability Criterion 113, Measurable Sets 114

3.5 Predictable and Previsible Processes ...................... 115

Predictable Processes 115,

Predictable Stopping

Times 118, Accessible Stopping Times 122

3.6 Special Properties of Daniell’s Mean ...................... 123

Maximality 123, Continuity Along Increasing Sequences 124, Predictable

Envelopes 125, Regularity 128, Stability Under Change of Measure 129

3.7 The Indenite Integral .................................... 130

The Indenite Integral 132, Integration Theory of the Indenite Integral 135,

A General Integrability Criterion 137, Approximation of the Integral via Parti-

tions 138, Pathwise Computation of the Indenite Integral 140, Integrators of

Finite Variation 144

3.8 Functions of Integrators .................................. 145

Square Bracket and Square Function of an Integrator 148, The Square Bracket of

Two Integrators 150, The Square Bracket of an Indenite Integral 153, Application:

The Jump of an Indenite Integral 155

3.9 Ito’s Formula ............................................. 157

The Doleans–Dade Exponential 159, Additional Exercises 161, Girsanov Theo-

rems 162, The Stratonovich Integral 168

3.10 Random Measures ........................................ 171

-Additivity 174, Law and Canonical Representation 175, Example: Wiener

Random Measure 177, Example: The Jump Measure of an Integrator 180, Strict

Random Measures and Point Processes 183, Example: Poisson Point Processes 184,

The Girsanov Theorem for Poisson Point Processes 185

Chapter 4 Control of Integral and Integrator ..................... 187

4.1 Change of Measure — Factorization ...................... 187

A Simple Case 187, The Main Factorization Theorem 191, Proof for p > 0 195,

Proof for p = 0 205

4.2 Martingale Inequalities ................................... 209

Feerman’s Inequality 209, The Burkholder–Davis–Gundy Inequalities 213, The

Hardy Mean 216, Martingale Representation on Wiener Space 218, Additional

Exercises 219

4.3 The Doob–Meyer Decomposition ......................... 221

Doleans–Dade Measures and Processes 222, Proof of Theorem 4.3.1: Necessity,

Uniqueness, and Existence 225, Proof of Theorem 4.3.1: The Inequalities 227, The

Previsible Square Function 228, The Doob–Meyer Decomposition of a Random

Measure 231

4.4 Semimartingales .......................................... 232

Integrators Are Semimartingales 233, Various Decompositions of an Integrator 234

4.5 Previsible Control of Integrators .......................... 238

Controlling a Single Integrator 239, Previsible Control of Vectors of Integrators 246,

Previsible Control of Random Measures 251

4.6 Levy Processes ........................................... 253

The Levy–Khintchine Formula 257, The Martingale Representation Theorem 261,

Canonical Components of a Levy Process 265, Construction of Levy Processes 267,

Feller Semigroup and Generator 268

Previsible Processes 118,

Contents

Chapter 5 Stochastic Dierential Equations ....................... 271

5.1 Introduction .............................................. 271

First Assumptions on the Data and Denition of Solution 272, Example: The

Ordinary Dierential Equation (ODE) 273, ODE: Flows and Actions 278, ODE:

Approximation 280

5.2 Existence and Uniqueness of the Solution ................. 282

The Picard Norms 283, Lipschitz Conditions 285, Existence and Uniqueness

of the Solution 289, Stability 293, Dierential Equations Driven by Random

Measures 296, The Classical SDE 297

5.3 Stability: Dierentiability in Parameters .................. 298

The Derivative of the Solution 301, Pathwise Dierentiability 303, Higher Order

Derivatives 305

5.4 Pathwise Computation of the Solution .................... 310

The Case of Markovian Coupling Coe cients 311, The Case of Endogenous Cou-

pling Coe cients 314, The Universal Solution 316, A Non-Adaptive Scheme 317,

The Stratonovich Equation 320, Higher Order Approximation: Obstructions 321,

Higher Order Approximation: Results 326

5.5 Weak Solutions ........................................... 330

The Size of the Solution 332, Existence of Weak Solutions 333, Uniqueness 337

5.6 Stochastic Flows ......................................... 343

Stochastic Flows with a Continuous Driver 343, Drivers with Small Jumps 346,

Markovian Stochastic Flows 347, Markovian Stochastic Flows Driven by a Levy

Process 349

5.7 Semigroups, Markov Processes, and PDE ................. 351

Stochastic Representation of Feller Semigroups 351

Appendix A Complements to Topology and Measure Theory ...... 363

A.1 Notations and Conventions ............................... 363

A.2 Topological Miscellanea ................................... 366

The Theorem of Stone–Weierstraß 366, Topologies, Filters, Uniformities 373, Semi-

continuity 376, Separable Metric Spaces 377, Topological Vector Spaces 379, The

Minimax Theorem, Lemmas of Gronwall and Kolmogoro 382, Dierentiation 388

A.3 Measure and Integration .................................. 391

-Algebras 391, Sequential Closure 391, Measures and Integrals 394, Order-

Continuous and Tight Elementary Integrals 398, Projective Systems of Mea-

sures 401, Products of Elementary Integrals 402, Innite Products of Elementary

Integrals 404, Images, Law, and Distribution 405, The Vector Lattice of All Mea-

sures 406, Conditional Expectation 407, Numerical and -Finite Measures 408,

Characteristic Functions 409, Convolution 413, Liftings, Disintegration of Mea-

sures 414, Gaussian and Poisson Random Variables 419

A.4 Weak Convergence of Measures ........................... 421

Uniform Tightness 425, Application: Donsker’s Theorem 426

A.5 Analytic Sets and Capacity ............................... 432

Applications to Stochastic Analysis 436, Supplements and Additional Exercises 440

A.6 Suslin Spaces and Tightness of Measures .................. 440

Polish and Suslin Spaces 440

A.7 The Skorohod Topology .................................. 443

A.8 The L p -Spaces ........................................... 448

Marcinkiewicz Interpolation 453, Khintchine’s Inequalities 455, Stable Type 458

Contents

A.9 Semigroups of Operators ................................. 463

Resolvent and Generator 463, Feller Semigroups 465, The Natural Extension of a

Feller Semigroup 467

Appendix B Answers to Selected Problems ....................... 470

References ....................................................... 477

Index of Notations ................................................ 483

Index ............................................................ 489

Answers .......... http://www.ma.utexas.edu/users/cup/Answers

Full Indexes ....... http://www.ma.utexas.edu/users/cup/Indexes

Errata ............. http://www.ma.utexas.edu/users/cup/Errata

Preface

This book originated with several courses given at the University of Texas.

The audience consisted of graduate students of mathematics, physics, electri-

cal engineering, and nance. Most had met some stochastic analysis during

work in their eld; the course was meant to provide the mathematical un-

derpinning. To satisfy the economists, driving processes other than Wiener

process had to be treated; to give the mathematicians a chance to connect

with the literature and discrete-time martingales, I chose to include driving

terms with jumps. This plus a predilection for generality for simplicity’s sake

led directly to the most general stochastic Lebesgue–Stieltjes integral.

The spirit of the exposition is as follows: just as having nite variation and

being right-continuous identies the useful Lebesgue–Stieltjes distribution

functions among all functions on the line, are there criteria for processes to

be useful as “random distribution functions.” They turn out to be straight-

forward generalizations of those on the line. A process that meets these

criteria is called an integrator , and its integration theory is just as easy as

that of a deterministic distribution function on the line – provided Daniell’s

method is used. (This proviso has to do with the lack of convexity in some

of the target spaces of the stochastic integral.)

For the purpose of error estimates in approximations both to the stochastic

integral and to solutions of stochastic dierential equations we dene various

numerical sizes of an integrator Z and analyze rather carefully how they

propagate through many operations done on and with Z , for instance, solving

a stochastic dierential equation driven by Z . These size-measurements

arise as generalizations to integrators of the famed Burkholder–Davis–Gundy

inequalities for martingales. The present exposition diers in the ubiquitous

use of numerical estimates from the many ne books on the market, where

convergence arguments are usually done in probability or every once in a

while in Hilbert space L 2 . For reasons that unfold with the story we employ

the L p -norms in the whole range 0 p < 1 . An eort is made to furnish

reasonable estimates for the universal constants that occur in this context.

Such attention to estimates, unusual as it may be for a book on this subject,

pays handsomely with some new results that may be edifying even to the

expert. For instance, it turns out that every integrator Z can be controlled

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