Sample-continuous process

In mathematics, a sample continuous process is a stochastic process whose sample paths are almost surely continuous functions.

Definition

Let $(\Omega ,{\mathcal {F}},\mathbb {P} )$ be a probability space. Let $X:I\times \Omega \to \mathbb {X}$ be a stochastic process, where the index set $I$ and state space $\mathbb {X}$ are both topological spaces. The process $X$ is called sample continuous (or almost surely continuous, or simply continuous) if the map $X(\omega ):I\to \mathbb {X}$ is continuous as a function of topological spaces for $\mathbb {P}$-almost all $\omega \in \Omega$.

In many examples, the index set is an interval of time, $[0,T]$ or $[0,+\infty )$, and the state space is the real line or $n$-dimensional Euclidean space.

Examples

• 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.

Properties

• For sample-continuous processes, the finite-dimensional distributions determine the law, and vice versa.