Let (Ω, Σ, P) be a probability space. Let X : I × Ω → S be a stochastic process, where the index set I and state space S are both topological spaces. Then the process X is called sample-continuous (or almost surely continuous, or simply continuous) if the map X(ω) : I → S is continuous as a function of topological spaces for P-almost all ω in Ω.
- Brownian motion (the Wiener process) on Euclidean space is sample-continuous.
- For "nice" parameters of the equations, solutions to stochastic differential equations are sample-continuous. See the existence and uniqueness theorem in the stochastic differential equations article for some sufficient conditions to ensure sample continuity.
- The process X : [0, +∞) × Ω → R that makes equiprobable jumps up or down every unit time according to
- is not sample-continuous. In fact, it is surely discontinuous.
- For sample-continuous processes, the finite-dimensional distributions determine the law, and vice versa.