Sample continuous process

From Wikipedia, the free encyclopedia

In mathematics, a sample continuous process is a stochastic process whose sample paths are almost surely continuous functions.

[edit] Definition

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $X : I \times \Omega \to \mathbb{X}$ be a stochastic process, where the index set $I$ and state space $\mathbb{X}$ are both topological spaces. The process $X$ is called sample continuous (or almost surely continuous, or simply continuous) if the map $X (\omega) : I \to \mathbb{X}$ is continuous as a function of topological spaces for $\mathbb{P}$ -almost all $\omega \in \Omega$ .

In many examples, the index set is an interval of time, $[0, T]$ or $[0, + \infty)$ , and the state space is the real line or $n$ -dimensional Euclidean space.

[edit] 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.

[edit] Properties

For sample-continuous processes, the finite-dimensional distributions determine the law, and vice versa.

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Category: Stochastic processes

Sample continuous process

From Wikipedia, the free encyclopedia

[edit] Definition

[edit] Examples

[edit] Properties

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