Stochastic Differential Equations

Abstract

Abstract We give an introduction to Brownian motion and illustrate that although sample paths are continuous, they are nowhere differentiable. This leads to a discussion on white and colored noise. In Section 3.3, we introduce stochastic integration and look at the difference between Itô and Stratonovich integrals. We examine Itô stochastic differential equations (SDEs) in Section 3.4, giving some motivating examples, and use the Itô formula to find some exact solutions. We then consider Stratonovich SDEs and show how to convert between Itô and Stratonovich cases. We discuss numerical approximation of both Itô and Stratonovich SDEs and discuss the multilevel Monte Carlo method for weak approximation in Section 3.6. We conclude in Section 3.7 by looking at the link between SDEs and partial differential equations (PDEs) and introduce both the Fokker–Planck and backward Fokker–Planck equations. Throughout, the aim is to develop intuition by first examining simple, one‐dimensional examples before giving more general formulations and results. Also included are some basic algorithms for numerical approximation.