Sample-continuous process

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

Definition
Let (&Omega;, &Sigma;, P) be a probability space. Let X : I &times; &Omega; &rarr; 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(&omega;) : I &rarr; S is continuous as a function of topological spaces for P-almost all &omega; in &Omega;.

In many examples, the index set I is an interval of time, [0, T] or [0, +&infin;), and the state space S is the real line or n-dimensional Euclidean space Rn.

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. See the existence and uniqueness theorem in the stochastic differential equations article for some sufficient conditions to ensure sample continuity.
 * The process X : [0, +&infin;) &times; &Omega; &rarr; R that makes equiprobable jumps up or down every unit time according to


 * $$\begin{cases} X_{t} \sim \mathrm{Unif} (\{X_{t-1} - 1, X_{t-1} + 1\}), & t \mbox{ an integer;} \\ X_{t} = X_{\lfloor t \rfloor}, & t \mbox{ not an integer;} \end{cases}$$


 * is not sample-continuous. In fact, it is surely discontinuous.

Properties

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