Table of Contents
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
1.1l 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 model189
Chapter 12 Example in Finance: Black-Scholes Model 195
12.1 Introduction: what is an option? 195
12.2 Self-financing strategies 197
12.2.1 Portfolio and its trading strategy 197
12.2.2 Self-financing strategies 198
12.2.3 Stock discount 200
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
