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In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
relating to
stochastic processes, a Feller process is a particular kind of
Markov process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
.
Definitions
Let ''X'' be a
locally compact Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
with a
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
base. Let ''C''
0(''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 ''C''
0(''X'') with the sup norm is a
Banach space.
A Feller semigroup on ''C''
0(''X'') is a collection
''t'' ≥ 0 of positive
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s from ''C''
0(''X'') to itself such that
* , , ''T''
''t''''f'' , , ≤ , , ''f'' , , for all ''t'' ≥ 0 and ''f'' in ''C''
0(''X''), i.e., it is a
contraction (in the weak sense);
* the
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
property: ''T''
''t'' + ''s'' = ''T''
''t'' o''T''
''s'' for all ''s'', ''t'' ≥ 0;
* lim
''t'' → 0, , ''T''
''t''''f'' − ''f'' , , = 0 for every ''f'' in ''C''
0(''X''). Using the semigroup property, this is equivalent to the map ''T''
''t''''f'' from ''t'' in
,∞)_to__''C''0(''X'')_being_right_continuous_for_every_''f''.
Warning:_This_terminology_is_not_uniform_across_the_literature._In_particular,_the_assumption_that_''T''
''t''_maps_''C''
0(''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_
A_Markov_chain_or_Markov_process_is_a__stochastic_model_describing_a_sequence_of_possible_events_in_which_the_probability_of_each_event_depends_only_on_the_state_attained_in_the_previous_event._Informally,_this_may_be_thought_of_as,_"What_happe_...
_with_a_Feller_transition_function.
__Generator_
Feller_processes_(or_transition_semigroups)_can_be_described_by_their_infinitesimal_generator_(stochastic_processes).html" ;"title="right_continuous.html" ;"title=",∞) to ''C''
0(''X'') being right continuous">,∞) to ''C''
0(''X'') being right continuous for every ''f''.
Warning: This terminology is not uniform across the literature. In particular, the assumption that ''T''
''t'' maps ''C''
0(''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
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
with a Feller transition function.
Generator
Feller processes (or transition semigroups) can be described by their infinitesimal generator (stochastic processes)">infinitesimal generator. A function ''f'' in ''C''
0 is said to be in the domain of the generator if the uniform limit
:
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 ''C''
0(''X'') to itself defined by
:
It can be shown that it satisfies the identity
:
Furthermore, for any fixed ''λ'' > 0, the image of ''R
λ'' is equal to the domain ''D
A'' of the generator ''A'', and
:
Examples
*
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
and the
Poisson process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
are examples of Feller processes. More generally, every
Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which disp ...
is a Feller process.
*
Bessel process In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process.
Formal definition
The Bessel process of order ''n'' is the real-valued process ''X'' given (when ''n'' ≥ 2) by
:X_t = \, W_t \, ,
whe ...
es are Feller processes.
* Solutions to
stochastic differential equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
s with
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...
coefficients are Feller processes.
* Every adapted right continuous Feller process on probability space
- satisfies the
strong Markov property with respect to the filtration
, i.e. for each
-
stopping time
In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time ) is a specific type of “random time”: a random variable whose value is inter ...
, conditioned on the event
, we have that for each
,
is independent of
given
.
[Rogers, L.C.G. and Williams, David ''Diffusions, Markov Processes and Martingales'' volume One: Foundations, second edition, John Wiley and Sons Ltd, 1979. (page 247, Theorem 8.3) ]
See also
*
Markov process
A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
*
Markov chain
*
Hunt process
*
Infinitesimal generator (stochastic processes)
In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a Feller process (i.e. a continuous-time Markov process satisfying certain regularity conditions) is a Fourier multiplier operator that encodes a g ...
References
{{DEFAULTSORT:Feller Process
Markov processes