# Sample-continuous process

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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 **R**^{n}.

## 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

- 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

## 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.