Stochastic Processes in General

Abstract

Let (Ω, F, P) be a probability space, (E, e) a measurable space, and Σ an arbitrary set. A stochastic process is a family X = { X t : t ∈ Σ} of random variables X t : Ω ↦ E. The space (E, e) is refered to as the state space of X. Given a point w in the sample space Ω the mapping t ↦ X t (ω) is called a trajectory or realization or sample path of the process X. For the general theory of stochastic processes see Dellacherie and Meyer (1976, 1980, 1983, 1987) and Gikhman and Skorokhod (1974, 1975, 1976).