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 $ X = ( x _ {t} , {\mathcal F} _ {t} , {\mathsf P} _ {x} ) $ with time shift operators $ \theta _ {t} $ is given on a measurable space $ ( E, {\mathcal B} ) $, let $ {\mathcal N} $ be the smallest $ \sigma $- algebra in the space of elementary events containing every event of the form $ \{ \omega : {x _ {t} \in B } \} $, where $ t \geq 0 $, $ B \in {\mathcal B} $, and let $ \overline{ {\mathcal N} }\; $ be the intersection of all completions of $ {\mathcal N} $ by all possible measures $ {\mathsf P} _ {x} $( $ x \in E $). A random function $ \gamma _ {t} $, $ t \geq 0 $, is called a functional of the Markov process $ X $ if, for every $ t \geq 0 $, $ \gamma _ {t} $ is measurable relative to the $ \sigma $- algebra $ \overline{ {\mathcal N} }\; _ {t} \cap {\mathcal F} _ {t} $.

Of particular interest are multiplicative and additive functionals of Markov processes. The first of these are distinguished by the condition $ \gamma _ {t + s } = \gamma _ {t} \theta _ {t} \gamma _ {s} $, and the second by the condition $ \gamma _ {t + s } = \gamma _ {t} + \theta _ {t} \gamma _ {s} $, $ s, t \geq 0 $, where $ \gamma _ {t} $ is assumed to be continuous on the right on $ [ 0, \infty ) $( on the other hand, it is sometimes appropriate to assume that these conditions are satisfied only $ {\mathsf P} _ {x} $- almost certainly for all fixed $ s, t \geq 0 $). One takes analogous formulations in the case of stopped and inhomogeneous processes. One can obtain examples of additive functionals of a Markov process $ X = ( x _ {t} , \zeta , {\mathcal F} _ {t} , {\mathsf P} ) $ by setting $ \gamma _ {t} $ for $ t < \zeta $ equal to $ f ( x _ {t} ) - f ( x _ {0} ) $, or to $ \int _ {0} ^ {t} f ( x _ {s} ) ds $, or to the sum of the jumps of the random function $ f ( x _ {s} ) $ for $ s \in [ 0, t] $, where $ f ( x) $ is bounded and measurable relative to $ {\mathcal B} $( the second and third examples are only valid under certain additional restrictions). Passing from any additive functional $ \gamma _ {t} $ to $ \mathop{\rm exp} \gamma _ {t} $ 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 $ t < \tau $ and to 0 for $ t \geq \tau $, where $ \tau $ is the first exit moment of $ X $ from some set $ A \in {\mathcal B} $, that is, $ \tau = \inf \{ {t \in [ 0, \zeta ] } : {x _ {t} \notin A } \} $.

There is a natural transformation of a Markov process — passage to a subprocess — associated with multiplicative functionals, subject to the condition $ 0 \leq \gamma _ {t} \leq 1 $. From the transition function $ {\mathsf P} ( t, x, B) $ of the process $ X = ( x _ {t} , \zeta , {\mathcal F} _ {t} , {\mathsf P} _ {x} ) $ one constructs a new one,

$$ \widetilde {\mathsf P} ( t, x, B) = \ \int\limits _ {\{ x _ {t} \in B \} } \gamma _ {t} {\mathsf P} _ {x} \{ d \omega \} ,\ \ A \in {\mathcal B} , $$

where it can happen that $ \widetilde {\mathsf P} ( 0, x, E) < 1 $ for certain points $ x \in E $. The new transition function in $ ( E, {\mathcal B} ) $ corresponds to some Markov process $ \widetilde{X} = ( \widetilde{x} _ {t} , \widetilde \zeta , {\mathcal F} tilde _ {t} , {\mathsf P} _ {x} ) $, which can be realized together with the original process on one and the same space of elementary events with the same measures $ {\mathsf P} _ {x} $, $ x \in E $, and, moreover, such that $ \widetilde \zeta \leq \zeta $, $ \widetilde{x} _ {t} = x _ {t} $ for $ 0 \leq t < \widetilde \zeta $ and such that the $ \sigma $- algebra $ {\mathcal F} tilde _ {t} $ is the trace of $ {\mathcal F} _ {t} $ in the set $ \{ \omega : {\widetilde \zeta > t } \} $. The process $ \widetilde{X} $ is called the subprocess of the Markov process $ X $ 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 $ X $ on the set $ A $; its phase space is naturally taken to be not the whole of $ ( E, {\mathcal B} ) $, but only $ ( A, {\mathcal B} _ {A} ) $, where $ {\mathcal B} _ {A} = \{ {B \in {\mathcal B} } : {B \subset A } \} $.

Additive functionals $ \gamma _ {t} \geq 0 $ 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 $ \gamma _ {t} \geq 0 $ is a continuous additive functional of a standard Markov process $ X $, with $ \gamma _ {t} > 0 $ for $ t > 0 $. Then $ Y = ( X _ {\tau _ {t} } , \gamma _ {\zeta ^ {-} } , {\mathcal F} _ {\tau _ {t} } , {\mathsf P} _ {x} ) $ is a standard Markov process, where $ \tau _ {t} = \sup \{ {s } : {\gamma _ {m} \leq t } \} $ for $ t \in [ 0, \gamma _ {\zeta ^ {-} } ) $. Here one says that $ Y $ is obtained from $ X $ as a result of the random change $ t \rightarrow \tau _ {t} $.

Various classes of additive functionals have been well studied, mainly of standard processes.

References

[LS]	R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , 1–2 , Springer (1977–1978) (Translated from Russian) MR1800858 MR1800857 MR0608221 MR0488267 MR0474486 Zbl 1008.62073 Zbl 1008.62072 Zbl 0556.60003 Zbl 0369.60001 Zbl 0364.60004
[D]	E.B. Dynkin, "Foundations of the theory of Markov processes" , Springer (1961) (Translated from Russian) MR0131898
[D2]	E.B. Dynkin, "Markov processes" , 1–2 , Springer (1965) (Translated from Russian) MR0193671 Zbl 0132.37901
[R]	D. Revuz, "Mesures associees aux fonctionelles additive de Markov I" Trans. Amer. Math. Soc. , 148 (1970) pp. 501–531
[B]	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 MR0415781 MR0415782

Comments

The trace of an algebra of sets $ {\mathcal F} $ in $ \Omega $ with respect to a subset $ \Omega ^ \prime \subset \Omega $ is the algebra of sets $ \Omega \cap {\mathcal F} = \{ {A \cap \Omega } : {A \in {\mathcal F} } \} $. It is a $ \sigma $- algebra if $ {\mathcal F} $ is a $ \sigma $- algebra.