To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time. 4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds # 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.