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.

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 1.11. 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