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''Markov time; stopping time''
 
''Markov time; stopping time''
  
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[[Category:Stochastic processes]]
 
[[Category:Stochastic processes]]
  
A notion used in probability theory for random variables having the property of independence of the "future" . More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624801.png" /> be a [[Measurable space|measurable space]] with a non-decreasing family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624803.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624804.png" />-algebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624805.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624806.png" /> in the case of continuous time and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624807.png" /> in the case of discrete time). A [[Random variable|random variable]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624808.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m0624809.png" /> is called a Markov moment or Markov time (relative to the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248011.png" />) if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248012.png" /> the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248013.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248014.png" />. In the case of discrete time this is equivalent to saying that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248015.png" /> the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248016.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248017.png" />.
+
A notion used in probability theory for random variables having the property of independence of the "future" . More precisely, let $  ( \Omega , {\mathcal F} ) $
 +
be a [[Measurable space|measurable space]] with a non-decreasing family $  ( {\mathcal F} _ {t} ) $,  
 +
$  t \in T $,  
 +
of $  \sigma $-
 +
algebras of $  {\mathcal F} $(
 +
$  T = [ 0 , \infty ] $
 +
in the case of continuous time and $  T = \{ 0 , 1 ,\dots \} $
 +
in the case of discrete time). A [[Random variable|random variable]] $  \tau = \tau ( \omega ) $
 +
with values in $  T \cup \{ + \infty \} $
 +
is called a Markov moment or Markov time (relative to the family $  ( {\mathcal F} _ {t} ) $,  
 +
$  t \in T $)  
 +
if for each $  t \in T $
 +
the event $  \{ \tau ( \omega ) \leq  t \} $
 +
belongs to $  {\mathcal F} _ {t} $.  
 +
In the case of discrete time this is equivalent to saying that for any $  n \in \{ 0 , 1 ,\dots \} $
 +
the event $  \{ \tau ( \omega ) = n \} $
 +
belongs to $  {\mathcal F} _ {n} $.
  
 
===Examples.===
 
===Examples.===
  
 +
1) Let  $  X ( t) $,
 +
$  t \in T $,
 +
be a real-valued right-continuous random process given on  $  ( \Omega , {\mathcal F} ) $,
 +
and let  $  {\mathcal F} _ {t} = \sigma \{  \omega  : {X ( s),  s \leq  t } \} $.
 +
Then the random variables
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248019.png" />, be a real-valued right-continuous random process given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248020.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248021.png" />. Then the random variables
+
$$
 
+
\tau ( \omega ) = \inf  \{ {t \geq  0 } : {X ( t) \in B } \}
<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/m/m062/m062480/m06248022.png" /></td> </tr></table>
+
$$
  
 
and
 
and
  
<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/m/m062/m062480/m06248023.png" /></td> </tr></table>
+
$$
 +
\sigma ( \omega )  = \inf \{ {t > 0 } : {X ( t) \in B } \}
 +
,
 +
$$
  
that is, the (first and first after <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248024.png" />) times of hitting the (Borel) set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248025.png" />, form Markov moments (in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248026.png" /> it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248027.png" />).
+
that is, the (first and first after $  + 0 $)  
 +
times of hitting the (Borel) set $  B $,  
 +
form Markov moments (in the case $  \{ \cdot \} = \emptyset $
 +
it is assumed that $  \inf  \emptyset = \infty $).
  
2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248029.png" />, is a standard [[Wiener process|Wiener process]], then the Markov moment
+
2) If $  w ( t) $,  
 +
$  t \geq  0 $,  
 +
is a standard [[Wiener process|Wiener process]], then the Markov moment
  
<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/m/m062/m062480/m06248030.png" /></td> </tr></table>
+
$$
 +
\tau  = \inf  \{ {t \geq  0 } : {w ( t) \geq  a } \}
 +
,\  a > 0 ,
 +
$$
  
 
has probability density
 
has probability density
  
<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/m/m062/m062480/m06248031.png" /></td> </tr></table>
+
$$
 +
P ( t)  = \
 +
 
 +
\frac{a}{t  ^ {3/2} \sqrt {2 \pi } }
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248032.png" />, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248033.png" />.
+
e ^ {- a  ^ {2} / 2 t } .
 +
$$
 +
 
 +
Here $  {\mathsf P} \{ \tau < \infty \} = 1 $,  
 +
but $  {\mathsf E} \tau = \infty $.
  
 
3) The random variable
 
3) The random variable
  
<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/m/m062/m062480/m06248034.png" /></td> </tr></table>
+
$$
 +
\gamma  = \inf  \{ {t > 0 } : {X ( s) \in B , s \geq  t } \}
 +
,
 +
$$
  
being the first time after which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248035.png" /> remains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062480/m06248036.png" />, is an example of a non-Markov moment (a random variable depending on the "future" ).
+
being the first time after which $  X _ {t} $
 +
remains in $  B $,  
 +
is an example of a non-Markov moment (a random variable depending on the "future" ).
  
 
Using the idea of a Markov moment one can formulate the strong [[Markov property|Markov property]] of Markov processes (cf. [[Markov process|Markov process]]). Markov moments and stopping times (that is, finite Markov moments) play a major role in the general theory of random processes and statistical [[Sequential analysis|sequential analysis]].
 
Using the idea of a Markov moment one can formulate the strong [[Markov property|Markov property]] of Markov processes (cf. [[Markov process|Markov process]]). Markov moments and stopping times (that is, finite Markov moments) play a major role in the general theory of random processes and statistical [[Sequential analysis|sequential analysis]].
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====Comments====
 
====Comments====
 
  
 
====References====
 
====References====

Latest revision as of 07:59, 6 June 2020


Markov time; stopping time

2020 Mathematics Subject Classification: Primary: 60G40 [MSN][ZBL]

A notion used in probability theory for random variables having the property of independence of the "future" . More precisely, let $ ( \Omega , {\mathcal F} ) $ be a measurable space with a non-decreasing family $ ( {\mathcal F} _ {t} ) $, $ t \in T $, of $ \sigma $- algebras of $ {\mathcal F} $( $ T = [ 0 , \infty ] $ in the case of continuous time and $ T = \{ 0 , 1 ,\dots \} $ in the case of discrete time). A random variable $ \tau = \tau ( \omega ) $ with values in $ T \cup \{ + \infty \} $ is called a Markov moment or Markov time (relative to the family $ ( {\mathcal F} _ {t} ) $, $ t \in T $) if for each $ t \in T $ the event $ \{ \tau ( \omega ) \leq t \} $ belongs to $ {\mathcal F} _ {t} $. In the case of discrete time this is equivalent to saying that for any $ n \in \{ 0 , 1 ,\dots \} $ the event $ \{ \tau ( \omega ) = n \} $ belongs to $ {\mathcal F} _ {n} $.

Examples.

1) Let $ X ( t) $, $ t \in T $, be a real-valued right-continuous random process given on $ ( \Omega , {\mathcal F} ) $, and let $ {\mathcal F} _ {t} = \sigma \{ \omega : {X ( s), s \leq t } \} $. Then the random variables

$$ \tau ( \omega ) = \inf \{ {t \geq 0 } : {X ( t) \in B } \} $$

and

$$ \sigma ( \omega ) = \inf \{ {t > 0 } : {X ( t) \in B } \} , $$

that is, the (first and first after $ + 0 $) times of hitting the (Borel) set $ B $, form Markov moments (in the case $ \{ \cdot \} = \emptyset $ it is assumed that $ \inf \emptyset = \infty $).

2) If $ w ( t) $, $ t \geq 0 $, is a standard Wiener process, then the Markov moment

$$ \tau = \inf \{ {t \geq 0 } : {w ( t) \geq a } \} ,\ a > 0 , $$

has probability density

$$ P ( t) = \ \frac{a}{t ^ {3/2} \sqrt {2 \pi } } e ^ {- a ^ {2} / 2 t } . $$

Here $ {\mathsf P} \{ \tau < \infty \} = 1 $, but $ {\mathsf E} \tau = \infty $.

3) The random variable

$$ \gamma = \inf \{ {t > 0 } : {X ( s) \in B , s \geq t } \} , $$

being the first time after which $ X _ {t} $ remains in $ B $, is an example of a non-Markov moment (a random variable depending on the "future" ).

Using the idea of a Markov moment one can formulate the strong Markov property of Markov processes (cf. Markov process). Markov moments and stopping times (that is, finite Markov moments) play a major role in the general theory of random processes and statistical sequential analysis.

References

[GS] I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027

Comments

References

[BG] R.M. Blumenthal, R.K. Getoor, "Markov processes and potential theory" , Acad. Press (1968) MR0264757 Zbl 0169.49204
[Do] J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984) pp. 390 MR0731258 Zbl 0549.31001
[Dy] E.B. Dynkin, "Markov processes" , 1 , Springer (1965) (Translated from Russian) MR0193671 Zbl 0132.37901
[W] A.D. Wentzell, "A course in the theory of stochastic processes" , McGraw-Hill (1981) (Translated from Russian) MR0781738 MR0614594 Zbl 0502.60001
[B] L.P. Breiman, "Probability" , Addison-Wesley (1968) MR0229267 Zbl 0174.48801
How to Cite This Entry:
Markov moment. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_moment&oldid=26569
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article