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.