Functional of a Markov process

2020 Mathematics Subject Classification: Primary: 60Jxx Secondary: 60J5560J57 [MSN][ZBL]

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,

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.