Functional of a Markov process
A random variable or random function depending in a measurable way on the trajectory of the Markov process; the condition of measurability varies according to the concrete situation. In the general theory of Markov processes one takes the following definition of a functional. Suppose that a non-stopped homogeneous Markov process with time shift operators
is given on a measurable space
, let
be the smallest
-algebra in the space of elementary events containing every event of the form
, where
,
, and let
be the intersection of all completions of
by all possible measures
(
). A random function
,
, is called a functional of the Markov process
if, for every
,
is measurable relative to the
-algebra
.
Of particular interest are multiplicative and additive functionals of Markov processes. The first of these are distinguished by the condition , and the second by the condition
,
, where
is assumed to be continuous on the right on
(on the other hand, it is sometimes appropriate to assume that these conditions are satisfied only
-almost certainly for all fixed
). One takes analogous formulations in the case of stopped and inhomogeneous processes. One can obtain examples of additive functionals of a Markov process
by setting
for
equal to
, or to
, or to the sum of the jumps of the random function
for
, where
is bounded and measurable relative to
(the second and third examples are only valid under certain additional restrictions). Passing from any additive functional
to
provides an example of a multiplicative functional. In the case of a standard Markov process, an interesting and important example of a multiplicative functional is given by the random function that is equal to 1 for
and to 0 for
, where
is the first exit moment of
from some set
, that is,
.
There is a natural transformation of a Markov process — passage to a subprocess — associated with multiplicative functionals, subject to the condition . From the transition function
of the process
one constructs a new one,
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where it can happen that for certain points
. The new transition function in
corresponds to some Markov process
, which can be realized together with the original process on one and the same space of elementary events with the same measures
,
, and, moreover, such that
,
for
and such that the
-algebra
is the trace of
in the set
. The process
is called the subprocess of the Markov process
obtained as a result of "killing" or shortening the lifetime. The subprocess corresponding to the multiplicative functional in the previous example is called the part of
on the set
; its phase space is naturally taken to be not the whole of
, but only
, where
.
Additive functionals give rise to another type of transformation of Markov processes — a random time change — which reduces to changing the time of traversing the various sections of a trajectory. Suppose, for example, that
is a continuous additive functional of a standard Markov process
, with
for
. Then
is a standard Markov process, where
for
. Here one says that
is obtained from
as a result of the random change
.
Various classes of additive functionals have been well studied, mainly of standard processes.
References
[1] | R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , 1–2 , Springer (1977–1978) (Translated from Russian) |
[2] | E.B. Dynkin, "Foundations of the theory of Markov processes" , Springer (1961) (Translated from Russian) |
[3] | E.B. Dynkin, "Markov processes" , 1–2 , Springer (1965) (Translated from Russian) |
[4] | D. Revuz, "Mesures associees aux fonctionelles additive de Markov I" Trans. Amer. Math. Soc. , 148 (1970) pp. 501–531 |
[5] | A. Benveniste, "Application de deux théorèmes de G. Mokobodzki à l'étude du noyau de Lévy d'un processus de Hunt sans hypothèse (L)" , Lect. notes in math. , 321 , Springer (1973) pp. 1–24 |
Comments
The trace of an algebra of sets in
with respect to a subset
is the algebra of sets
. It is a
-algebra if
is a
-algebra.
Functional of a Markov process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Functional_of_a_Markov_process&oldid=21161