Wikipedia / Feller process
Wikipedia / Feller process
In probability theory relating to stochastic processes, a Feller process is a particular kind of Markov process.
Definitions
be a locally compact Hausdorff space with a countable base denote the space of all real-valued continuous functions that vanish at infinity, equipped with the sup-norm with the sup norm is a Banach space.
A Feller semigroup is a contraction C0-semigroup of positive operators of linear maps to itself with the following properties:
is a positive operator;
, i.e., it is a contraction (in the weak sense);
, i.e., it is a semigroup;
being right continuous.
Warning, 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 is said to be in the domain of the generator if the uniform limit
.
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 to itself defined by
It can be shown that it satisfies the identity
, and
Examples
Brownian motion and the Poisson process). 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 satisfies the strong Markov property-stopping time.
See also
References