Feller process

Short description: Stochastic process

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

Definitions

Let X be a locally compact Hausdorff 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 ||. From analysis, we know that C₀(X) with the sup norm is a Banach space.

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

• ||T_tf || ≤ ||f || for all t ≥ 0 and f in C₀(X), i.e., it is a contraction (in the weak sense);
• the semigroup property: T_t + s = T_t ∘T_s for all s, t ≥ 0;
• lim_t → 0||T_tf − f || = 0 for every f in C₀(X). Using the semigroup property, this is equivalent to the map T_tf  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 T_t maps C₀(X) into itself is replaced by some authors by the condition that it maps C_b(X), the space of bounded continuous functions, into itself. The reason for this is twofold: first, it allows including 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

[math]\displaystyle{ Af = \lim_{t\rightarrow 0} \frac{T_tf - f}{t}, }[/math]

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

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 (R_λ)_λ > 0 from C₀(X) to itself defined by

[math]\displaystyle{ R_\lambda f = \int_0^\infty e^{-\lambda t}T_t f\,dt. }[/math]

It can be shown that it satisfies the identity

[math]\displaystyle{ R_\lambda R_\mu = R_\mu R_\lambda = (R_\mu-R_\lambda)/(\lambda-\mu). }[/math]

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

[math]\displaystyle{ \begin{align} & R_\lambda = (\lambda - A)^{-1}, \\ & A = \lambda - R_\lambda^{-1}. \end{align} }[/math]

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 adapted right continuous Feller process on a filtered probability space [math]\displaystyle{ (\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\geq 0}) }[/math] satisfies the strong Markov property with respect to the filtration [math]\displaystyle{ (\mathcal{F}_{t^+})_{t\geq 0} }[/math], i.e., for each [math]\displaystyle{ (\mathcal{F}_{t^+})_{t\geq 0} }[/math]-stopping time [math]\displaystyle{ \tau }[/math], conditioned on the event [math]\displaystyle{ \{\tau \lt \infty\} }[/math], we have that for each [math]\displaystyle{ t\ge 0 }[/math], [math]\displaystyle{ X_{\tau + t} }[/math] is independent of [math]\displaystyle{ \mathcal{F}_{\tau^+} }[/math] given [math]\displaystyle{ X_\tau }[/math].

See also

• Markov process
• Markov chain
• Hunt process
• Infinitesimal generator (stochastic processes)