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====References====
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , '''1–2''' , Springer (1977–1978) (Translated from Russian) {{MR|1800858}} {{MR|1800857}} {{MR|0608221}} {{MR|0488267}} {{MR|0474486}} {{ZBL|1008.62073}} {{ZBL|1008.62072}} {{ZBL|0556.60003}} {{ZBL|0369.60001}} {{ZBL|0364.60004}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> E.B. Dynkin, "Foundations of the theory of Markov processes" , Springer (1961) (Translated from Russian) {{MR|0131898}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.B. Dynkin, "Markov processes" , '''1–2''' , Springer (1965) (Translated from Russian) {{MR|0193671}} {{ZBL|0132.37901}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D. Revuz, "Mesures associees aux fonctionelles additive de Markov I" ''Trans. Amer. Math. Soc.'' , '''148''' (1970) pp. 501–531</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> 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 {{MR|0415781}} {{MR|0415782}} {{ZBL|}} </TD></TR></table>
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|valign="top"|{{Ref|LS}}|| R.S. Liptser, A.N. Shiryaev, "Statistics of random processes" , '''1–2''' , Springer (1977–1978) (Translated from Russian) {{MR|1800858}} {{MR|1800857}} {{MR|0608221}} {{MR|0488267}} {{MR|0474486}} {{ZBL|1008.62073}} {{ZBL|1008.62072}} {{ZBL|0556.60003}} {{ZBL|0369.60001}} {{ZBL|0364.60004}}
 
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|valign="top"|{{Ref|D}}|| E.B. Dynkin, "Foundations of the theory of Markov processes" , Springer (1961) (Translated from Russian) {{MR|0131898}} {{ZBL|}}
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|valign="top"|{{Ref|D2}}|| E.B. Dynkin, "Markov processes" , '''1–2''' , Springer (1965) (Translated from Russian) {{MR|0193671}} {{ZBL|0132.37901}}
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|valign="top"|{{Ref|R}}|| D. Revuz, "Mesures associees aux fonctionelles additive de Markov I" ''Trans. Amer. Math. Soc.'' , '''148''' (1970) pp. 501–531
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|valign="top"|{{Ref|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 {{MR|0415781}} {{MR|0415782}} {{ZBL|}}
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====Comments====
 
====Comments====
 
The trace of an algebra of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208079.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208080.png" /> with respect to a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208081.png" /> is the algebra of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208082.png" />. It is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208083.png" />-algebra if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208084.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208085.png" />-algebra.
 
The trace of an algebra of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208079.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208080.png" /> with respect to a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208081.png" /> is the algebra of sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208082.png" />. It is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208083.png" />-algebra if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208084.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208085.png" />-algebra.

Revision as of 06:29, 13 May 2012

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

[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 in with respect to a subset is the algebra of sets . It is a -algebra if is a -algebra.

How to Cite This Entry:
Functional of a Markov process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Functional_of_a_Markov_process&oldid=26521
This article was adapted from an original article by M.G. Shur (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article