Feller-continuous process
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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.
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Definition[edit]
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[edit]
- 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 Itō diffusion with Lipschitz-continuous drift and diffusion coefficients is a Feller-continuous process.
See also[edit]
References[edit]
- Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Lemma 8.1.4)