Introduction to Stochastic Analysis Integrals and Differential Equations

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ISBN: 9781848213111 | 1848213115
Cover: Hardcover
Copyright: 8/15/2011

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This book provides an introduction to stochastic analysis in an easy-to-understand format implied for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. It includes information on 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, and simulations of solutions of SDEs (strong and weak approximations). Exercises with hints and/or solutions are included as well.

Preface	p. 9
Notation	p. 13
Introduction: Basic Notions of Probability Theory	p. 17
Probability space	p. 17
Random variables	p. 21
Characteristics of a random variable	p. 21
Types of random variables	p. 23
Conditional probabilities and distributions	p. 26
Conditional expectations as random variables	p. 27
Independent events and random variables	p. 29
Convergence of random variables	p. 29
Cauchy criterion	p. 31
Series of random variables	p. 31
Lebesgue theorem	p. 32
Fubini theorem	p. 32
Random processes	p. 33
Kolmogorov theorem	p. 34
Brownian Motion	p. 35
Definition and properties	p. 35
White noise and Brownian motion	p. 45
Exercises	p. 49
Stochastic Models with Brownian Motion and White Noise	p. 51
Discrete time	p. 51
Continuous time	p. 55
Stochastic Integral with Respect to Brownian Motion	p. 59
Preliminaries Stochastic integral with respect to a step process	p. 59
Definition and properties	p. 69
Extensions	p. 81
Exercises	p. 85
Itô's Formula	p. 87
Exercises	p. 94
Stochastic Differential Equations	p. 97
Exercises	p. 105
Itô Processes	p. 107
Exercises	p. 121
Stratonovich Integral and Equations	p. 125
Exercises	p. 136
Linear Stochastic Differential Equations	p. 137
Explicit solution of a linear SDE	p. 137
Expectation and variance of a solution of an LSDE	p. 141
Other explicitly solvable equations	p. 145
Stochastic exponential equation	p. 147
Exercises	p. 153
Solutions of SDEs as Markov Diffusion Processes	p. 155
Introduction	p. 155
Backward and forward Kolmogorov equations	p. 161
Stationary density of a diffusion process	p. 172
Exercises	p. 176
Examples	p. 179
Additive noise: Langevin equation	p. 180
Additive noise: general case	p. 180
Multiplicative noise: general remarks	p. 184
Multiplicative noise: Verhulst equation	p. 186
Multiplicative noise: genetic model189
Example in Finance: Black-Scholes Model	p. 195
Introduction: what is an option?	p. 195
Self-financing strategies	p. 197
Portfolio and its trading strategy	p. 197
Self-financing strategies	p. 198
Stock discount	p. 200
Option pricing problem: the Black-Scholes model	p. 204
Black-Scholes formula	p. 206
Risk-neutral probabilities: alternative derivation of Black-Scholes formula	p. 210
Exercises	p. 214
Numerical Solution of Stochastic Differential Equations	p. 217
Memories of approximations of ordinary differential equations	p. 218
Euler approximation	p. 221
Higher-order strong approximations	p. 224
First-order weak approximations	p. 231
Higher-order weak approximations	p. 238
Example: Milstein-type approximations	p. 241
Example: Runge-Kutta approximations	p. 244
Exercises	p. 249
Elements of Multidimensional Stochastic Analysis	p. 251
Multidimensional Brownian motion	p. 251
Itô's formula for a multidimensional Brownian motion	p. 252
Stochastic differential equations	p. 253
Itô processes	p. 254
Itô's formula for multidimensional Itô processes	p. 256
Linear stochastic differential equations	p. 256
Diffusion processes	p. 257
Approximations of stochastic differential equations	p. 259
Solutions, Hints, and Answers	p. 261
Bibliography	p. 271
Index	p. 273
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Introduction to Stochastic Analysis Integrals and Differential Equations

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