Ebook: Introduction to Stochastic Analysis

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 naive 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; Ito and Stratonovich stochastic integrals, Ito’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.Content:
Chapter 1 Introduction (pages 17–34):
Chapter 2 Brownian Motion (pages 35–50):
Chapter 3 Stochastic Models with Brownian Motion and White Noise (pages 51–57):
Chapter 4 Stochastic Integral with Respect to Brownian Motion (pages 59–86):
Chapter 5 Ito's Formula (pages 87–95):
Chapter 6 Stochastic Differential Equations (pages 97–105):
Chapter 7 Ito Processes (pages 107–123):
Chapter 8 Stratonovich Integral and Equations (pages 125–136):
Chapter 9 Linear Stochastic Differential Equations (pages 137–154):
Chapter 10 Solutions of SDEs as Markov Diffusion Processes (pages 155–177):
Chapter 11 Examples (pages 179–193):
Chapter 12 Example in Finance (pages 195–215):
Chapter 13 Numerical Solution of Stochastic Differential Equations (pages 217–250):
Chapter 14 Elements of Multidimensional Stochastic Analysis (pages 251–259):