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. Higher-order strong approximations
- 13.4. First-order weak approximations
- 13.5. Higher-order weak approximations
- 13.6. Example: Milstein-type 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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