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,
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.
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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://www.encyclopediaofmath.org/index.php?title=Functional_of_a_Markov_process&oldid=26521