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
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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, structure, space, models, and change. History On ...
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 Cro ...
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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_ , +∞) × 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_(stochastic_processes)">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_vario_...
_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_function.html" ;"title="Law_(stochastic_processes).html" "title="probability_space.html" ;"title=", +∞) × Ω → R''n'', defined on a probability space">, +∞) × 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 (stochastic processes)">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 vario ...
 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 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 In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0 ...
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 of ...
with Lipschitz-continuous drift and diffusion coefficients is a Feller-continuous process.


See also

* Continuous stochastic process


References

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