Stochastic Processes Defined by ODE’s

Abstract

A stochastic process in the time interval [0,∞) is defined as a family of random variables X t (ω), t ≥ 0, ω ∈ Ω, on a measurable space (Ω,F, P). To describe the probability structure of the process, one should define the family of distributions of (Xt1,…, Xtn) for any integer n and any 0 ≤ t 1 < t 2 < … < tn. This family of finite dimensional distributions is, in general, a rather bulky subject. Therefore, as a rule, special classes of stochastic processes are considered for which such a description can be reduced to more convenient characteristics. First, we consider the basic processes which have a simple statistical structure. Then we consider relatively simple and explicitly defined transformations of these basic processes. The Wiener process, Poisson process, and continuous time Markov chains with finite number of states will be our basic processes. We assume that the main properties of these processes are known. Actually, all classes of continuous time stochastic processes, allowing deep enough theory, can be constructed from these basic processes using relatively simple transformations.