Feller process
Encyclopedia
In probability theory
relating to stochastic processes, a Feller process is a particular kind of Markov process
.
with a countable
base
. Let C0(X) denote the space of all real-valued continuous function
s 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
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.
. A function f in C0 is said to be in the domain of the generator if the uniform limit
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.
It can be shown that it satisfies the identity
Furthermore, for any fixed λ > 0, the image of Rλ is equal to the domain DA of the generator A, and
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
relating to stochastic processes, a Feller process is a particular kind of Markov process
Markov process
In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...
.
Definitions
Let X be a locally compact topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
with a countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
base
Base (topology)
In mathematics, a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T...
. Let C0(X) denote the space of all real-valued continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s 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 contractionContraction mappingIn mathematics, a contraction mapping, or contraction, on a metric space is a function f from M to itself, with the property that there is some nonnegative real number k...
(in the weak sense); - the semigroupSemigroupIn mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
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 generatorInfinitesimal generator
In mathematics, the term infinitesimal generator may refer to:* an element of the Lie algebra associated to a Lie group;* the infinitesimal generator of a stochastic process;* the infinitesimal generator of a strongly continuous semigroup....
. A function f in C0 is said to be in the domain of the generator if the uniform limit
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 byIt can be shown that it satisfies the identity
Furthermore, for any fixed λ > 0, the image of Rλ is equal to the domain DA of the generator A, and
Examples
- Brownian motion and the Poisson process are examples of Feller processes. More generally, every Lévy processLévy processIn probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below...
is a Feller process.
- Bessel processBessel processIn mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process. The n-dimensional Bessel process is the real-valued process X given byX_t = \| W_t \|,...
es are Feller processes.
- Solutions to stochastic differential equationStochastic differential equationA stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....
s with Lipschitz continuous coefficients are Feller processes.
- Every Feller process satisfies the strong Markov property.
See also
- Markov processMarkov processIn probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...
- Markov chainMarkov chainA Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...
- Hunt process
- Infinitesimal generator (stochastic processes)Infinitesimal generator (stochastic processes)In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a stochastic process is a partial differential operator that encodes a great deal of information about the process...