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{{MSC|60Jxx|60J55,60J57}}
 
{{MSC|60Jxx|60J55,60J57}}
  
 
[[Category:Markov processes]]
 
[[Category:Markov processes]]
  
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|Markov process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f0420801.png" /> with time shift operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f0420802.png" /> is given on a [[Measurable space|measurable space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f0420803.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f0420804.png" /> be the smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f0420805.png" />-algebra in the space of elementary events containing every event of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f0420806.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f0420807.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f0420808.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f0420809.png" /> be the intersection of all completions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208010.png" /> by all possible measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208012.png" />). A random function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208014.png" />, is called a functional of the Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208015.png" /> if, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208017.png" /> is measurable relative to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208018.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208019.png" />.
+
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|Markov process]] $  X = ( x _ {t} , {\mathcal F} _ {t} , {\mathsf P} _ {x} ) $
 +
with time shift operators $  \theta _ {t} $
 +
is given on a [[Measurable space|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208020.png" />, and the second by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208023.png" /> is assumed to be continuous on the right on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208024.png" /> (on the other hand, it is sometimes appropriate to assume that these conditions are satisfied only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208025.png" />-almost certainly for all fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208026.png" />). One takes analogous formulations in the case of stopped and inhomogeneous processes. One can obtain examples of additive functionals of a Markov process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208027.png" /> by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208028.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208029.png" /> equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208030.png" />, or to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208031.png" />, or to the sum of the jumps of the random function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208032.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208034.png" /> is bounded and measurable relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208035.png" /> (the second and third examples are only valid under certain additional restrictions). Passing from any additive functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208036.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208037.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208038.png" /> and to 0 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208040.png" /> is the first exit moment of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208041.png" /> from some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208042.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208043.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208044.png" />. From the transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208045.png" /> of the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042080/f04208046.png" /> one constructs a new one,
+
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,
  
<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>
+
$$
 +
\widetilde {\mathsf P}  ( t, x, B)  = \
 +
\int\limits _ {\{ x _ {t} \in B \} }
 +
\gamma _ {t} {\mathsf P} _ {x} \{ d \omega \} ,\ \
 +
A \in {\mathcal B} ,
 +
$$
  
where it can happen that <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 $  \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 <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 $  \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.
 
Various classes of additive functionals have been well studied, mainly of standard processes.
Line 31: Line 121:
  
 
====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 $  {\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.

Latest revision as of 19:40, 5 June 2020


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.

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=47016
This article was adapted from an original article by M.G. Shur (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article