Feller-continuous process
- Feller-continuous process
In mathematics, a Feller-continuous process is a continuous-time stochastic process satisfying a stronger continuity property than simple sample continuity. Intuitively, a Feller-continuous process is one for which the expected value of suitable statistics of the process depends continuously on the initial condition of the process. The concept is named after Croatian-American mathematician William Feller.
Definition
Let "X" : [0, +∞) × Ω → R"n", defined on a probability space (Ω, Σ, P), be a stochastic process. For a point "x" ∈ R"n", let P"x" denote the law of "X" given initial datum "X"0 = "x", and let E"x" denote expectation with respect to P"x". Then "X" is said to be a Feller-continuous process if, for any fixed "t" ≥ 0 and any bounded, continuous and Σ-measurable function "g" : R"n" → R, E"x" ["g"("X""t")] depends continuously upon "x".
Examples
* Any process "X" whose paths are almost surely constant for all time is a Feller-continuous process, since then E"x" ["g"("X""t")] is simply "g"("x"), which, by hypothesis, depends continuously upon "x".
* Any Itō diffusion with Lipschitz-continuous drift and diffusion coefficients is a Feller-continuous process.
ee also
* Continuous stochastic process
References
* cite book
last = Øksendal
first = Bernt K.
authorlink = Bernt Øksendal
title = Stochastic Differential Equations: An Introduction with Applications
edition = Sixth edition
publisher=Springer
location = Berlin
year = 2003
id = ISBN 3-540-04758-1 (See Lemma 8.1.4)
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