Martingales, Stopping Times, and Filtrations

Abstract

A stochastic process is a mathematical model for the occurrence, at each moment after the initial time, of a random phenomenon. The randomness is captured by the introduction of a measurable space (Ω, ℱ), called the sample space, on which probability measures can be placed. Thus, a stochastic process is a collection of random variables X = {X_t; 0≤t<∞} on (Ω, ℱ), which take values in a second measurable space (S, S ) called the state space. For our purposes, the state space (SS) will be the d-dimensional Euclidean space equipped with the σ-field of Borel sets, i.e., S= ℝ^d, S= ℬ (ℝ^d), where ℬ(U) will always be used to denote the smallest a-field containing all open sets of a topological space U. The index t ∈ [O, ∞) of the random variables X, admits a convenient interpretation as time.