Feller process

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Feller process" – news · newspapers · books · scholar · JSTOR (September 2009) (Learn how and when to remove this message)

In

Definitions

Let X be a

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

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

${\displaystyle Af=\lim _{t\rightarrow 0}{\frac {T_{t}f-f}{t}},}$

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

${\displaystyle R_{\lambda }f=\int _{0}^{\infty }e^{-\lambda t}T_{t}f\,dt.}$

It can be shown that it satisfies the identity

${\displaystyle R_{\lambda }R_{\mu }=R_{\mu }R_{\lambda }=(R_{\mu }-R_{\lambda })/(\lambda -\mu ).}$

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

${\displaystyle {\begin{aligned}&R_{\lambda }=(\lambda -A)^{-1},\\&A=\lambda -R_{\lambda }^{-1}.\end{aligned}}}$

Examples

• Poisson process are examples of Feller processes. More generally, every Lévy process
  is a Feller process.
• Bessel processes are Feller processes.
• Solutions to
  Lipschitz continuous coefficients are Feller processes.^{[citation needed}
  ]
• Every adapted right continuous Feller process on a filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0})}$ satisfies the
  strong Markov property
  with respect to the filtration ${\displaystyle ({\mathcal {F}}_{t^{+}})_{t\geq 0}}$, i.e., for each ${\displaystyle ({\mathcal {F}}_{t^{+}})_{t\geq 0}}$-stopping time ${\displaystyle \tau }$, conditioned on the event ${\displaystyle \{\tau <\infty \}}$, we have that for each ${\displaystyle t\geq 0}$, ${\displaystyle X_{\tau +t}}$ is independent of ${\displaystyle {\mathcal {F}}_{\tau ^{+}}}$ given ${\displaystyle X_{\tau }}$.^[1]

See also

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

References