Book description
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
 Cover
 Title Page
 Copyright
 Preface
 Notation

Chapter 1: Introduction: Basic Notions of Probability Theory
 1.1. Probability space
 1.2. Random variables
 1.3. Characteristics of a random variable
 1.4. Types of random variables
 1.5. Conditional probabilities and distributions
 1.6. Conditional expectations as random variables
 1.7. Independent events and random variables
 1.8. Convergence of random variables
 1.9. Cauchy criterion
 1.10. Series of random variables
 1.11. Lebesgue theorem
 1.12. Fubini theorem
 1.13. Random processes
 1.14. Kolmogorov theorem
 Chapter 2: Brownian Motion
 Chapter 3: Stochastic Models with Brownian Motion and White Noise
 Chapter 4: Stochastic Integral with Respect to Brownian Motion
 Chapter 5: Itô's Formula
 Chapter 6: Stochastic Differential Equations
 Chapter 7: Itô Processes
 Chapter 8: Stratonovich Integral and Equations
 Chapter 9: Linear Stochastic Differential Equations
 Chapter 10: Solutions of SDEs as Markov Diffusion Processes
 Chapter 11: Examples
 Chapter 12: Example in Finance: Black–Scholes Model

Chapter 13: Numerical Solution of Stochastic Differential Equations
 13.1. Memories of approximations of ordinary differential equations
 13.2. Euler approximation
 13.3. Higherorder strong approximations
 13.4. Firstorder weak approximations
 13.5. Higherorder weak approximations
 13.6. Example: Milsteintype approximations
 13.7. Example: Runge–Kutta approximations
 13.8. Exercises

Chapter 14: Elements of Multidimensional Stochastic Analysis
 14.1. Multidimensional Brownian motion
 14.2. Itô's formula for a multidimensional Brownian motion
 14.3. Stochastic differential equations
 14.4. Itô processes
 14.5. Itô's formula for multidimensional Itô processes
 14.6. Linear stochastic differential equations
 14.7. Diffusion processes
 14.8. Approximations of stochastic differential equations
 Solutions, Hints, and Answers
 Bibliography
 Index
Product information
 Title: Introduction to Stochastic Analysis: Integrals and Differential Equations
 Author(s):
 Release date: August 2011
 Publisher(s): Wiley
 ISBN: 9781848213111
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