Introduction to Stochastic Analysis

Introduction to Stochastic Analysis: Integrals and Differential Equations (ISTE)
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

Chapter 1: Introduction: Basic Notions of ProbabilityTheory 14
Chapter 2: Brownian Motion 32
Chapter 3: Stochastic Models with Brownian Motion andWhite Noise 48
Chapter 4: Stochastic Integral with Respect to BrownianMotion 55
Chapter 5: Itô’s Formula 83
Chapter 6: Stochastic Differential Equations 92
Chapter 7: Itô Processes 101
Chapter 8: Stratonovich Integral and Equations 118
Chapter 9: Linear Stochastic Differential Equations 130
Chapter 10: Solutions of SDEs as Markov DiffusionProcesses 148
Chapter 11: Examples 171
Chapter 12: Example in Finance: Black–Scholes Model 186
Chapter 13: Numerical Solution of Stochastic DifferentialEquations 207
Chapter 14: Elements of Multidimensional StochasticAnalysis 241