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.

References

• Kloeden, Peter E. (1992). Numerical solution of stochastic differential equations. Applications of Mathematics (New York) 23. Berlin: Springer-Verlag. pp. pp. 38–39, . ISBN 3-540-54062-8. {{cite book}}: |pages= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)CS1 maint: extra punctuation (link)