Sample-continuous process

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title: "Sample-continuous process" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["stochastic-processes"] topic_path: "general/stochastic-processes" source: "https://en.wikipedia.org/wiki/Sample-continuous_process" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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

References

  • {{cite book | author = Kloeden, Peter E. |author2=Platen, Eckhard | title = Numerical solution of stochastic differential equations | series = Applications of Mathematics (New York) 23 |publisher = Springer-Verlag | location = Berlin | year = 1992 | pages = 38–39 | isbn = 3-540-54062-8

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stochastic-processes