Feller-continuous process explained

Feller-continuous process should not be confused with Feller process.

In mathematics, a Feller-continuous process is a continuous-time stochastic process for which the expected value of suitable statistics of the process at a given time in the future depend 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 ∈ Rⁿ, let P^x denote the law of X given initial value X₀ = 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 : Rⁿ → R, E^x[''g''(''X''_''t'')] depends continuously upon x.

Examples

• Every 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.
• Every Itô diffusion with Lipschitz-continuous drift and diffusion coefficients is a Feller-continuous process.

See also

• Continuous stochastic process

References

• Book: Øksendal , Bernt K. . Bernt Øksendal

. Bernt Øksendal. Stochastic Differential Equations: An Introduction with Applications. Sixth. Springer. Berlin. 2003. 3-540-04758-1. (See Lemma 8.1.4)