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Introduction to Stochastic Analysis: Integrals and Differential Equations

ISBN: 978-1-84821-311-1
288 pages
August 2011, Wiley-ISTE
Introduction to Stochastic Analysis: Integrals and Differential Equations (1848213115) cover image
This is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion processes.
The topics covered include Brownian motion; motivation of stochastic models with Brownian motion; Itô and Stratonovich stochastic integrals, Itô’s formula; stochastic differential equations (SDEs); solutions of SDEs as Markov processes; application examples in physical sciences and finance; simulation of solutions of SDEs (strong and weak approximations). Exercises with hints and/or solutions are also provided.
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Preface 9

Notation 13

Chapter 1. Introduction: Basic Notions of Probability Theory 17

1.1. Probability space 17

1.2. Random variables 21

1.3. Characteristics of a random variable 21

1.4. Types of random variables 23

1.5. Conditional probabilities and distributions 26

1.6. Conditional expectations as random variables 27

1.7. Independent events and random variables 29

1.8. Convergence of random variables 29

1.9. Cauchy criterion 31

1.10. Series of random variables 31

1.11. Lebesgue theorem 32

1.12. Fubini theorem 32

1.13. Random processes 33

1.14. Kolmogorov theorem 34

Chapter 2. Brownian Motion 35

2.1. Definition and properties 35

2.2. White noise and Brownian motion 45

2.3. Exercises 49

Chapter 3. Stochastic Models with Brownian Motion and White Noise 51

3.1. Discrete time 51

3.2. Continuous time 55

Chapter 4. Stochastic Integral with Respect to Brownian Motion 59

4.1. Preliminaries. Stochastic integral with respect to a step process 59

4.2. Definition and properties 69

4.3. Extensions 81

4.4. Exercises 85

Chapter 5. Itô’s Formula 87

5.1. Exercises 94

Chapter 6. Stochastic Differential Equations 97

6.1. Exercises 105

Chapter 7. Itô Processes 107

7.1. Exercises 121

Chapter 8. Stratonovich Integral and Equations 125

8.1. Exercises 136

Chapter 9. Linear Stochastic Differential Equations 137

9.1. Explicit solution of a linear SDE 137

9.2. Expectation and variance of a solution of an LSDE 141

9.3. Other explicitly solvable equations 145

9.4. Stochastic exponential equation 147

9.5. Exercises 153

Chapter 10. Solutions of SDEs as Markov Diffusion Processes 155

10.1. Introduction 155

10.2. Backward and forward Kolmogorov equations 161

10.3. Stationary density of a diffusion process 172

10.4. Exercises 176

Chapter 11. Examples 179

11.1. Additive noise: Langevin equation 180

11.2. Additive noise: general case 180

11.3. Multiplicative noise: general remarks 184

11.4. Multiplicative noise: Verhulst equation 186

11.5. Multiplicative noise: genetic model 189

Chapter 12. Example in Finance: Black–Scholes Model 195

12.1. Introduction: what is an option? 195

12.2. Self-financing strategies 197

12.3. Option pricing problem: the Black–Scholes model 204

12.4. Black–Scholes formula 206

12.5. Risk-neutral probabilities: alternative derivation of Black–Scholes formula 210

12.6. Exercises 214

Chapter 13. Numerical Solution of Stochastic Differential Equations 217

13.1. Memories of approximations of ordinary differential equations  218

13.2. Euler approximation 221

13.3. Higher-order strong approximations 224

13.4. First-order weak approximations 231

13.5. Higher-order weak approximations 238

13.6. Example: Milstein-type approximations 241

13.7. Example: Runge–Kutta approximations 244

13.8. Exercises 249

Chapter 14. Elements of Multidimensional Stochastic Analysis 251

14.1. Multidimensional Brownian motion 251

14.2. Itô’s formula for a multidimensional Brownian motion 252

14.3. Stochastic differential equations 253

14.4. Itô processes 254

14.5. Itô’s formula for multidimensional Itô processes 256

14.6. Linear stochastic differential equations 256

14.7. Diffusion processes 257

14.8. Approximations of stochastic differential equations 259

Solutions, Hints, and Answers 261

Bibliography 271

Index 273

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“Thus, the book is a welcome addition in the effort to make stochastic integration and SDE as accessible as possible to the greater public interested in or in need of using them.”  (Mathematical  Reviews, 1 February 2013)

“If I have a chance to teach (again) a course in stochastic financial modelling, I will definitely choose this to be among two or three sources to use. I have all the reasons to strongly recommend it to anybody in the area of modern stochastic modelling.”  (Zentralblatt MATH, 1 December 2012)

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