Feller-continuous process

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Continuous-time stochastic process

Not to be confused with Feller 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.

Definition

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Let X : [0, +) x O - Rⁿ, defined on a probability space (O, S, P), be a stochastic process. For a point x Rⁿ, let P^x denote the law of X given initial value X₀ = 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 S-measurable function g : Rⁿ - R, E^x[g(X_t)] depends continuously upon x.

Examples

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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 Ito diffusion with Lipschitz-continuous drift and diffusion coefficients is a Feller-continuous process.

References

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Oksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Lemma 8.1.4)

v t e Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton-Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Additive process Airy process Bessel process Birth-death process pure birth Brownian motion Bridge Dyson Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming-Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Ito diffusion Ito process Jump diffusion Jump process Levy process Local time Markov additive process McKean-Vlasov process Ornstein-Uhlenbeck process Poisson process Compound Non-homogeneous Quasimartingale Schramm-Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman-Yor process Point process Cox Determinantal Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Binomial options pricing model Black-Derman-Toy Black-Karasinski Black-Scholes Chan-Karolyi-Longstaff-Sanders (CKLS) Chen Constant elasticity of variance (CEV) Cox-Ingersoll-Ross (CIR) Garman-Kohlhagen Heath-Jarrow-Morton (HJM) Heston Ho-Lee Hull-White Korn-Kreer-Lenssen LIBOR market Rendleman-Bartter SABR volatility Vasicek Wilkie
Actuarial models	Buhlmann Cramer-Lundberg Risk process Sparre-Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Cadlag paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss-Markov Markov Mixing Piecewise-deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher-Tippett-Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero-one laws (Blumenthal, Borel-Cantelli, Engelbert-Schmidt, Hewitt-Savage, Kolmogorov, Levy)
Inequalities	Burkholder-Davis-Gundy Doob's martingale Doob's upcrossing Kunita-Watanabe Marcinkiewicz-Zygmund
Tools	Cameron-Martin theorem Convergence of random variables Doleans-Dade exponential Doob decomposition theorem Doob-Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman-Kac formula Filtration Girsanov theorem Infinitesimal generator Ito integral Ito's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Kosambi-Karhunen-Loeve theorem Levy-Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
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