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Feller-continuous process
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<h2 id="definition">Definition</h2> <p>Let <i>X</i> : [0, +∞) × Ω → R<i>n</i>, defined on a <a href="/facts/Probability_space/dEcv2VrU">probability space</a> (Ω, Σ, P), be a stochastic process. For a point <i>x</i> ∈ R<i>n</i>, let P<i>x</i> denote the <a href="/facts/Law_(stochastic_processes)/oprLtu0K">law</a> of <i>X</i> given initial value <i>X</i>0 = <i>x</i>, and let E<i>x</i> denote expectation with respect to P<i>x</i>. Then <i>X</i> is said to be a Feller-continuous process if, for any fixed <i>t</i> ≥ 0 and any <a href="/facts/Bounded_function/whVjbtuk">bounded</a>, continuous and Σ-<a href="/facts/Measurable_function/NgIHnb1b">measurable function</a> <i>g</i> : R<i>n</i> → R, E<i>x</i>[<i>g</i>(<i>X</i><i>t</i>)] depends continuously upon <i>x</i>. </p> <h2 id="examples">Examples</h2> <ul><li>Every process <i>X</i> whose paths are <a href="/facts/Almost_surely/fWaxzpBA">almost surely</a> constant for all time is a Feller-continuous process, since then E<i>x</i>[<i>g</i>(<i>X</i><i>t</i>)] is simply <i>g</i>(<i>x</i>), which, by hypothesis, depends continuously upon <i>x</i>.</li> <li>Every <a href="/facts/It%25C3%25B4_diffusion/vlwAqWSU">Itô diffusion</a> with <a href="/facts/Lipschitz_continuity/Hw20EPEU">Lipschitz-continuous</a> drift and diffusion coefficients is a Feller-continuous process.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Continuous_stochastic_process/GEmGTXYm">Continuous stochastic process</a></li></ul> <ul><li><a href="/facts/Bernt_%25C3%2598ksendal/Au4oSbnG">Øksendal, Bernt K.</a> (2003). <i>Stochastic Differential Equations: An Introduction with Applications</i> (Sixth ed.). Berlin: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-04758-1. (See Lemma 8.1.4)</li></ul>