Sample-continuous process
In mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions.
Definition
Let 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 if the map X : 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, or 0, +∞), and the state space S is the [real line or n-dimensional Euclidean space Rn.
Examples
- Brownian motion 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
Properties
- For sample-continuous processes, the [finite-dimensional distributions determine the law, and vice versa.