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Feller process

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In probability theory relating to stochastic processes, a Feller process is a particular kind of Markov process.

Contents

Definitions

Let X be a locally compact topological space with a countable base. Let C0(X) denote the space of all real-valued continuous functions on X that vanish at infinity, equipped with the sup-norm ||f ||.

A Feller semigroup on C0(X) is a collection {Tt}t ≥ 0 of positive linear maps from C0(X) to itself such that

  • ||Ttf || ≤ ||f || for all t ≥ 0 and f in C0(X), i.e., it is a contraction (in the weak sense);
  • the semigroup property: Tt + s = Tt oTs for all s, t ≥ 0;
  • limt → 0||Ttf − f || = 0 for every f in C0(X). Using the semigroup property, this is equivalent to the map Ttf  from t in [0,∞) to C0(X) being right continuous for every f.

    • Warning: This terminology is not uniform across the literature. In particular, the assumption that Tt maps C0(X) into itself is replaced by some authors by the condition that it maps Cb(X), the space of bounded continuous functions, into itself. The reason for this is twofold: first, it allows to include processes that enter "from infinity" in finite time. Second, it is more suitable to the treatment of spaces that are not locally compact and for which the notion of "vanishing at infinity" makes no sense.

    A Feller transition function is a probability transition function associated with a Feller semigroup.

    A Feller process is a Markov process with a Feller transition function.

    Generator

    Feller processes (or transition semigroups) can be described by their infinitesimal generator. A function f in C0 is said to be in the domain of the generator if the uniform limit

    A f = lim t → 0 T t f − f t ,

    exists. The operator A is the generator of Tt, and the space of functions on which it is defined is written as DA.

    A characterization of operators that can occur as the infinitesimal generator of Feller processes is given by the Hille-Yosida theorem. This uses the resolvent of the Feller semigroup, defined below.

    Resolvent

    The resolvent of a Feller process (or semigroup) is a collection of maps (Rλ)λ > 0 from C0(X) to itself defined by

    R λ f = ∫ 0 ∞ e − λ t T t f d t .

    It can be shown that it satisfies the identity

    R λ R μ = R μ R λ = ( R μ − R λ ) / ( λ − μ ) .

    Furthermore, for any fixed λ > 0, the image of Rλ is equal to the domain DA of the generator A, and

    R λ = ( λ − A ) − 1 , A = λ − R λ − 1 .

    Examples

  • Brownian motion and the Poisson process are examples of Feller processes. More generally, every Lévy process is a Feller process.
  • Bessel processes are Feller processes.
  • Solutions to stochastic differential equations with Lipschitz continuous coefficients are Feller processes.
  • Every Feller process satisfies the strong Markov property.

    • References

    Feller process Wikipedia
    (Text) CC BY-SA


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