title: Feller-continuous process
text: # Feller-continuous process
  
  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.
  
  ## Definition
  
  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 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 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 with Lipschitz-continuous drift and diffusion coefficients is a Feller-continuous process.
  