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.
1. Introduction: Basic Notions of Probability Theory.
2. Brownian Motion.
3. Stochastic Models with Brownian Motion and White Noise.
4. Stochastic Integral with Respect to Brownian Motion.
5. Itô’s Formula.
6. Stochastic Differential Equations.
7. Itô Processes.
8. Stratonovich Integral and Equations.
9. Linear Stochastic Differential Equations.
10. Solutions of SDEs as Markov Diffusion Processes.
12. Example in Finance: Black–Scholes Model.
13. Numerical Solution of Stochastic Differential Equations.
14. Elements of Multidimensional Stochastic Analysis.
Vigirdas Mackevicius is Professor of the Department of Mathematical Analysis in the Faculty of Mathematics of Vilnius University in Lithuania. His research interests include stochastic analysis, limit theorems for stochastic processes, and stochastic numerics.
Table of Contents
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