Wikipedia / Sample-continuous process
Wikipedia / Sample-continuous process
In mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions.
Definition
Let (,,P) be a probability space. Let X:IS 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():IS is continuous as a function of topological spaces for P-almost all in .
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,+)R that makes equiprobable jumps up or down every unit time according to
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.