# Sample continuous process

- 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 **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

::$egin\{cases\}\; X\_\{t\}\; sim\; mathrm\{Unif\}\; (\{X\_\{t-1\}\; -\; 1,\; X\_\{t-1\}\; +\; 1\}),\; t\; mbox\{\; an\; integer;\}\; \backslash \; X\_\{t\}\; =\; X\_\{lfloor\; t\; floor\},\; t\; mbox\{\; not\; an\; integer;\}\; end\{cases\}$

: 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.

**ee also**

* Continuous stochastic process

**References**

* cite book

author = Kloeden, Peter E.

coauthors = Platen, Eckhard

title = Numerical solution of stochastic differential equations

series = Applications of Mathematics (New York) 23

publisher = Springer-Verlag

location = Berlin

year = 1992

pages = pp. 38–39;

isbn = 3-540-54062-8

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