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

Properties

See also

References

  • Kloeden, Peter E.; Platen, Eckhard (1992). Numerical solution of stochastic differential equations. Applications of Mathematics (New York) 23. Berlin: Springer-Verlag. pp. 38–39. ISBN 3-540-54062-8.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.