Sample-continuous process

In mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions. In mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions. 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. In many examples, the index set I is an interval of time, or [0, +∞), and the state space S is the real line or n-dimensional Euclidean space Rn.