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Introduction to Stochastic Integration

Hui-Hsiung Kuo
Publisher: 
Springer Verlag
Publication Date: 
2006
Number of Pages: 
278
Format: 
Paperback
Series: 
Universitext
Price: 
49.95
ISBN: 
0-387-28720-5
Category: 
Monograph
[Reviewed by
Ita Cirovic Donev
, on
06/13/2006
]

Introduction to Stochastic Integration is exactly what the title says. I would maybe just add a “friendly” introduction because of the clear presentation and flow of the contents. The first three chapters are an introduction to Brownian motion. Readers who have had stochastic processes these three chapters will serve as a little remainder. For others I believe there is enough here to allow them grasp the concepts firmly. The next three chapters deal with stochastic integral followed by the Itô formula and a complete chapter on the application of Itô formula. Chapter 9 deals with multiple Wiener–Itô integrals. Stochastic differential equations, given Brownian motion, are discussed in Chapter 10. The last chapter presents some examples and more detailed applications of the theory. The author does not give much detail when it comes to applications, but rather shows how to apply the specific theoretical concepts to certain problems in different fields.

Given its clear structure and composition, the book could be useful for a short course on stochastic integration. The concepts are easy to grasp (given, of course, the appropriate background knowledge). Problems are given in each chapter and naturally are proof-based. As the book is intended for a “fast” introductory course on stochastic integration, and possibly for self-study, including solutions to problems would have been an advantage.

The reader should be well equipped with knowledge in advanced calculus and basic graduate level probability theory.

Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She hold a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Random Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Definition of Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Simple Properties of Brownian Motion . . . . . . . . . . . . . . . . . . . . . 8

2.3 Wiener Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Conditional Expectation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Series Expansion of Wiener Integrals . . . . . . . . . . . . . . . . . . . . . . . 20

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Constructions of Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1 Wiener Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Borel–Cantelli Lemma and Chebyshev Inequality . . . . . . . . . . . . 25

3.3 Kolmogorov’s Extension and Continuity Theorems . . . . . . . . . . . 27

3.4 L´evy’s Interpolation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Filtrations for a Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.3 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4 Simple Examples of Stochastic Integrals . . . . . . . . . . . . . . . . . . . . 48

4.5 Doob Submartingale Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.6 Stochastic Processes Defined by Itˆo Integrals . . . . . . . . . . . . . . . . 52

4.7 Riemann Sums and Stochastic Integrals . . . . . . . . . . . . . . . . . . . . 57

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

xii Contents

5 An Extension of Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . 61

5.1 A Larger Class of Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2 A Key Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3 General Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.4 Stopping Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.5 Associated Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6 Stochastic Integrals for Martingales . . . . . . . . . . . . . . . . . . . . . . . 75

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.2 Poisson Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.3 Predictable Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.4 Doob–Meyer Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . 80

6.5 Martingales as Integrators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.6 Extension for Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7 The Itˆo Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7.1 Itˆo’s Formula in the Simplest Form . . . . . . . . . . . . . . . . . . . . . . . . 93

7.2 Proof of Itˆo’s Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.3 Itˆo’s Formula Slightly Generalized . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.4 Itˆo’s Formula in the General Form . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.5 Multidimensional Itˆo’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.6 Itˆo’s Formula for Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8 Applications of the Itˆo Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.1 Evaluation of Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.2 Decomposition and Compensators . . . . . . . . . . . . . . . . . . . . . . . . . 117

8.3 Stratonovich Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8.4 L´evy’s Characterization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.5 Multidimensional Brownian Motions . . . . . . . . . . . . . . . . . . . . . . . 129

8.6 Tanaka’s Formula and Local Time . . . . . . . . . . . . . . . . . . . . . . . . . 133

8.7 Exponential Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.8 Transformation of Probability Measures . . . . . . . . . . . . . . . . . . . . 138

8.9 Girsanov Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

9 Multiple Wiener–Itˆo Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9.1 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9.2 Double Wiener–Itˆo Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

9.3 Hermite Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

9.4 Homogeneous Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

9.5 Orthonormal Basis for Homogeneous Chaos . . . . . . . . . . . . . . . . . 164

9.6 Multiple Wiener–Itˆo Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Contents xiii

9.7 Wiener–Itˆo Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

9.8 Representation of Brownian Martingales . . . . . . . . . . . . . . . . . . . . 180

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

10 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 185

10.1 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

10.2 Bellman–Gronwall Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

10.3 Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . 190

10.4 Systems of Stochastic Differential Equations . . . . . . . . . . . . . . . . 196

10.5 Markov Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

10.6 Solutions of Stochastic Differential Equations . . . . . . . . . . . . . . . 203

10.7 Some Estimates for the Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 208

10.8 Diffusion Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

10.9 Semigroups and the Kolmogorov Equations . . . . . . . . . . . . . . . . . 216

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

11 Some Applications and Additional Topics . . . . . . . . . . . . . . . . . . 231

11.1 Linear Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . 231

11.2 Application to Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

11.3 Application to Filtering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

11.4 Feynman–Kac Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

11.5 Approximation of Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . 254

11.6 White Noise and Electric Circuits . . . . . . . . . . . . . . . . . . . . . . . . . 258

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Glossary of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

Tags: 
Probability
Stochastic Calculus