Continuous time process and Brownian motion

Consider a complete probability space (Ω,F ,P;F) equipped with the Þltration F = {Ft; 0 ≤ t <∞}. A stochastic process is a collection of random variables X = {Xt; 0 ≤ t <∞} where, for every t, Xt : Ω → Rd is a random variable. We assume the space Rd is equipped with the usual Borel σ-algebra B(Rd). Every Þxed ω ∈ Ω corresponds to a sample path (or, trajectory), that is, the function t 7→ Xt(ω) for t ≥ 0. DeÞnition: The process is said to be continuous if every sample path is continuous on [0,∞). The processes we discuss here are all continuous, even though almost all the results can be carried over to the more general RCLL processes (right continuous, with Þnite left-hand limit).