Stochastic differential equations are differential equations whose solutions are stochastic processes. They exhibit appealing mathematical properties that are useful in modeling uncertainties and noisy phenomena in many disciplines. This book is motivated by applications of stochastic differential equations in target tracking and medical technology and, in particular, their use in methodologies such as filtering, smoothing, parameter estimation, and machine learning. It builds an intuitive hands-on understanding of what stochastic differential equations are all about, but also covers the essentials of Itô calculus, the central theorems in the field, and such approximation schemes as stochastic Runge-Kutta. Greater emphasis is given to solution methods than to analysis of theoretical properties of the equations. The book's practical approach assumes only prior understanding of ordinary differential equations. The numerous worked examples and end-of-chapter exercises include application-driven derivations and computational assignments. MATLAB/Octave source code is available for download, promoting hands-on work with the methods.
1. Introduction
2. Some background on ordinary differential equations
3. Pragmatic introduction to stochastic differential equations
4. Ito calculus and stochastic differential equations
5. Probability distributions and statistics of SDEs
6. Statistics of linear stochastic differential equations
7. Useful theorems and formulas for SDEs
8. Numerical simulation of SDEs
9. Approximation of nonlinear SDEs
10. Filtering and smoothing theory
11. Parameter estimation in SDE models
12. Stochastic differential equations in machine learning
13. Epilogue.
