Stochastic Differential Equations

Abstract

The chapter begins with Section 4.1 in which motivational examples of random walks and stochastic phenomena in nature are presented. In Section 4.2 the concept of random processes is introduced in a more precise way. In Section 4.3 the concept of a Gaussian and Markov random process is developed. In Section 4.4 the important special case of white noise is defined. White noise is the driving force for all of the stochastic processes studied in this book. Other sections in this chapter define Ito and Stratonovich stochastic differential equations (SDEs), their properties and corresponding Fokker–Planck equations, which describe how probability densities associated with SDEs evolve over time. In particular, Section 4.7 examines the Fokker–Planck equation for a particular kind of SDE called an Ornstein–Uhlenbeck process. And Section 4.8 examines how SDEs and Fokker–Planck equations change their appearance when different coordinate systems are used. The main points that the reader should take away from this chapter are: Whereas a deterministic system of ordinary differential equations that satisfies certain conditions (i.e., the Lipschitz conditions) are guaranteed to have a unique solution for any given initial conditions, when random noise is introduced the resulting “stochastic differential equation” will not produce repeatable solutions. It is the ensemble behavior of the sample paths obtained from numerically solving a stochastic differential equation many times that is important. This ensemble behavior can be described either as a stochastic integral (of which there are two main types, called Itˆo and Stratonovich), or by using a partial differential equation akin to the diffusion equations studied in Chapter 2, which is called the Fokker–Planck (or forward Kolmogorov) equation. Two different forms of the Fokker–Planck equation exist, corresponding to the interpretation of the solution of a given SDE as being either an Itˆo or Stratonovich integral, and an analytical apparatus exists for converting between these forms. Multi-dimensional SDEs in Rn can be written in Cartesian or curvilinear coordinates, but care must be taken when converting between coordinate systems because the usual rules of multivariable calculus do not apply in some situations.