Sample-continuous process

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.

• 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

::

: is not sample-continuous. In fact, it is surely discontinuous.

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

• Continuous stochastic process

• {{cite book

| author = Kloeden, Peter E.

|author2=Platen, Eckhard

| title = Numerical solution of stochastic differential equations

| series = Applications of Mathematics (New York) 23

|publisher = Springer-Verlag

| location = Berlin

| year = 1992

| pages = 38–39

| isbn = 3-540-54062-8

}}

{{Stochastic processes}}

Category:Stochastic processes