A Crash Course for Stochastic Calculus

In this section we want to understand what does the notation ∫ t 0 XsdWs mean. 1.1 Stochastic Process Xt Given a probability space {Ω,F,P} and time index set T , a stochastic process (Xs)s∈T is a measurable mapping from Ω×T to some space. In our most interested setting, the value space is R or Rd, while the time index set is T = [0,∞) with Borel σ-algebra. Another common option for T is the integer set: T = {0, 1, . . . , }, which is used for discrete time processes. However discrete time processes are usually called a “chain”. X is usually seen as X(ω, t) ∈ R. Notation wise, people usually omit the appearance of ω and instead write X(t) or Xt. When we say a stochastic process has certain property, for example continuous, we mean for a.s. fixed ω, the process X(ω, s)s≥0 is continuous.