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→Rn, defined on a probability space (Ω,Σ,P), be a stochastic process. For a point x∈Rn, let Px denote the law of X given initial value X0=x, and let Ex denote expectation with respect to Px. 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, Ex[g(Xt)] depends continuously upon x.

Examples

• Every process X whose paths are almost surely constant for all time is a Feller-continuous process, since then Ex[g(Xt)] 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

• (See Lemma 8.1.4)