Namespaces
Variants
Actions

Difference between revisions of "Functional of a Markov process"

From Encyclopedia of Mathematics
Jump to: navigation, search
(MSC|60Jxx|60J55,60J57 Category:Markov processes)
m (MR/ZBL numbers added)
Line 11: Line 11:
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208047.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208047.png" /></td> </tr></table>
  
where it can happen that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208048.png" /> for certain points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208049.png" />. The new transition function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208050.png" /> corresponds to some Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208051.png" />, which can be realized together with the original process on one and the same space of elementary events with the same measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208053.png" />, and, moreover, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208055.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208056.png" /> and such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208057.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208058.png" /> is the trace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208059.png" /> in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208060.png" />. The process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208061.png" /> is called the subprocess of the Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208062.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208063.png" /> on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208064.png" />; its phase space is naturally taken to be not the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208065.png" />, but only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208066.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208067.png" />.
+
where it can happen that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208048.png" /> for certain points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208049.png" />. The new transition function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208050.png" /> corresponds to some Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208051.png" />, which can be realized together with the original process on one and the same space of elementary events with the same measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208053.png" />, and, moreover, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208055.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208056.png" /> and such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208057.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208058.png" /> is the trace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208059.png" /> in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208060.png" />. The process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208061.png" /> is called the subprocess of the Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208062.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208063.png" /> on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208064.png" />; its phase space is naturally taken to be not the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208065.png" />, but only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208066.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208067.png" />.
  
 
Additive functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208068.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208069.png" /> is a continuous additive functional of a standard Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208070.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208071.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208072.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208073.png" /> is a standard Markov process, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208074.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208075.png" />. Here one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208076.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208077.png" /> as a result of the random change <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208078.png" />.
 
Additive functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208068.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208069.png" /> is a continuous additive functional of a standard Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208070.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208071.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208072.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208073.png" /> is a standard Markov process, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208074.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208075.png" />. Here one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208076.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208077.png" /> as a result of the random change <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208078.png" />.
Line 18: Line 18:
  
 
====References====
 
====References====
<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)</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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E.B. Dynkin,   "Markov processes" , '''1–2''' , Springer (1965) (Translated from Russian)</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</TD></TR></table>
+
<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>
  
  

Revision as of 10:30, 27 March 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

[1] 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
[2] E.B. Dynkin, "Foundations of the theory of Markov processes" , Springer (1961) (Translated from Russian) MR0131898
[3] E.B. Dynkin, "Markov processes" , 1–2 , Springer (1965) (Translated from Russian) MR0193671 Zbl 0132.37901
[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 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=21161
This article was adapted from an original article by M.G. Shur (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article