Sample-continuous process

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

Definition

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, [0, T] or [0, +∞), 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, +∞) × Ω → 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.

See also

• Continuous stochastic process

References

• Kloeden, Peter E.; Platen, Eckhard (1992). Numerical solution of stochastic differential equations. Applications of Mathematics (New York) 23. Berlin: Springer-Verlag. pp. 38–39;. ISBN 3-540-54062-8.