Feller-continuous process

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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.

Contents 1 Definition 2 Examples 3 See also 4 References

[edit] Definition

Let X : [0, +∞) × Ω → Rⁿ, 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 datum 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.

[edit] 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.

[edit] See also

• Continuous stochastic process

[edit] References

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