4 Stochastic Differential Equations

Abstract

Publisher SummaryThis chapter presents a study of stochastic differential equations. It discusses the modeling of a certain stochastic process as the solution of a (stochastic) differential equation. This leads to definition of the Itô stochastic integral and proof of existence and uniqueness of solutions of stochastic differential equations. This is followed by a formal derivation of Itô's lemma, which is a fundamental tool in continuous nonlinear filter theory. The chapter introduces another stochastic integral, that of Stratonovich, and discusses its relationship with the Itô integral. The Itô integral can be used in the derivation of Kolmogorov's equation, also known as the Fokker-Planck equation. This equation describes the evolution of the transition probability density of the Markov process generated by the Itô equation. A Markov process whose transition probability density satisfies Kolmogorov's equation is called a diffusion process.