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Feller-continuous 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&nbsp;:&nbsp;[0,&nbsp;+∞)&nbsp;×&nbsp;Ω&nbsp;→&nbsp;R<sup>n</sup>, defined on a probability space (Ω,&nbsp;Σ,&nbsp;P), be a stochastic process. For a point x&nbsp;∈&nbsp;R<sup>n</sup>, let P<sup>x</sup> denote the law of X given initial value X<sub>0</sub>&nbsp;=&nbsp;x, and let E<sup>x</sup> denote expectation with respect to P<sup>x</sup>. Then X is said to be a Feller-continuous process if, for any fixed t&nbsp;≥&nbsp;0 and any bounded, continuous and Σ-measurable function g&nbsp;:&nbsp;R<sup>n</sup>&nbsp;→&nbsp;R, E<sup>x</sup>[g(X<sub>t</sub>)] 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<sup>x</sup>[g(X<sub>t</sub>)] 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

References

  • (See Lemma 8.1.4)