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In mathematics, a Feller-continuous process is a continuous-time stochastic process for which the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
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
Croatia , image_flag = Flag of Croatia.svg , image_coat = Coat of arms of Croatia.svg , anthem = "Lijepa naša domovino"("Our Beautiful Homeland") , image_map = , map_caption = , capit ...
n-American
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
William Feller William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian- American mathematician specializing in probability theory. Early life and education Feller was born in Zagreb to Ida Oemichen-Perc, a Croa ...
.


Definition

Let ''X'' :  , +∞) × 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 Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vari ...
of ''X'' given initial value ''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">bounded function">bounded, continuous and Σ-measurable function ''g'' : R''n'' → 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 In mathematics – specifically, in stochastic analysis – an Itô diffusion is a solution to a specific type of stochastic differential equation. That equation is similar to the Langevin equation used in physics to describe the Brownian motion o ...
with Lipschitz-continuous drift and diffusion coefficients is a Feller-continuous process.


See also

*
Continuous stochastic process In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index parameter. Continuity is a nice property for (the sample paths of) a process to have, ...


References

* (See Lemma 8.1.4) {{Stochastic processes Stochastic processes