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An equation of the form
 
An equation of the form
  
<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/k/k055/k055680/k0556801.png" /></td> </tr></table>
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$$
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P ( s , x ; u , \Gamma )  = \
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\int\limits _ { E }
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P ( s , x ; t , d y )
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P ( t , y ; u , \Gamma ) ,\ \
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s < t < u ,
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$$
  
that is, a condition imposed on the [[Transition function|transition function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055680/k0556802.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055680/k0556803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055680/k0556804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055680/k0556805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055680/k0556806.png" /> being a measurable space), enabling one (under certain conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055680/k0556807.png" />) to construct a [[Markov process|Markov process]] for which the conditional probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055680/k0556808.png" /> is the same as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055680/k0556809.png" />. Conversely, for a Markov process its transition function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055680/k05568010.png" />, which by definition is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055680/k05568011.png" />, satisfies the Kolmogorov–Chapman equation, as follows immediately from general properties of conditional probabilities. This was pointed out by S. Chapman [[#References|[1]]] and investigated by A.N. Kolmogorov in 1931 (see [[#References|[2]]]).
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that is, a condition imposed on the [[Transition function|transition function]] $  P ( s , x ;  t , \Gamma ) $(
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0 \leq  s \leq  t < \infty $,
 +
$  x \in E $,  
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$  \Gamma \in \mathfrak B $,  
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$  ( E , \mathfrak B ) $
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being a measurable space), enabling one (under certain conditions on $  ( E , \mathfrak B ) $)  
 +
to construct a [[Markov process|Markov process]] for which the conditional probability $  {\mathsf P} _ {s,x} ( x _ {t} \in \Gamma ) $
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is the same as $  P ( s , x ;  t , \Gamma ) $.  
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Conversely, for a Markov process its transition function $  P ( s , x ;  t , \Gamma ) $,  
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which by definition is equal to $  {\mathsf P} _ {s,x} ( x _ {t} \in \Gamma ) $,  
 +
satisfies the Kolmogorov–Chapman equation, as follows immediately from general properties of conditional probabilities. This was pointed out by S. Chapman {{Cite|C}} and investigated by A.N. Kolmogorov in 1931 (see {{Cite|K}}).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"S. Chapman,   ''Proc. Roy. Soc. Ser. A'' , '''119'''  (1928)  pp. 34–54</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"A. [A.N. Kolmogorov] Kolmogoroff,  "Ueber die analytischen Methoden in der Wahrscheinlichkeitsrechnung"  ''Math. Ann.'' , '''104'''  (1931)  pp. 415–458</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. [I.I. Gikhman] Gihman,  A.V. [A.V. Skorokhod] Skorohod,  "The theory of stochastic processes" , '''2''' , Springer  (1975)  (Translated from Russian)</TD></TR></table>
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{|
 
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|valign="top"|{{Ref|C}}|| S. Chapman, "?", ''Proc. Roy. Soc. Ser. A'' , '''119'''  (1928)  pp. 34–54
 
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|-
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|valign="top"|{{Ref|K}}|| A. Kolmogoroff,  "Ueber die analytischen Methoden in der Wahrscheinlichkeitsrechnung"  ''Math. Ann.'' , '''104'''  (1931)  pp. 415–458
 +
|-
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|valign="top"|{{Ref|GS}}|| I.I. Gihman,  A.V. Skorohod,  "The theory of stochastic processes" , '''2''' , Springer  (1975)  (Translated from Russian)
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|}
  
 
====Comments====
 
====Comments====
Line 20: Line 52:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Lévy,   "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.B. Dynkin,   "Markov processes" , '''1''' , Springer (1965) pp. Sect. 5.26 (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"W. Feller,   "An introduction to probability theory and its applications" , '''1''' , Wiley (1966) pp. Chapt. XV.13</TD></TR></table>
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{|
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|valign="top"|{{Ref|L}}|| P. Lévy, "Processus stochastiques et mouvement Brownien", Gauthier-Villars (1965)
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|-
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|valign="top"|{{Ref|D}}|| E.B. Dynkin, "Markov processes", '''1''', Springer (1965) pp. Sect. 5.26 (Translated from Russian)
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|-
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|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its  applications"]], '''1''', Wiley (1966) pp. Chapt. XV.13
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|}

Latest revision as of 22:14, 5 June 2020


2020 Mathematics Subject Classification: Primary: 60J35 [MSN][ZBL]

An equation of the form

$$ P ( s , x ; u , \Gamma ) = \ \int\limits _ { E } P ( s , x ; t , d y ) P ( t , y ; u , \Gamma ) ,\ \ s < t < u , $$

that is, a condition imposed on the transition function $ P ( s , x ; t , \Gamma ) $( $ 0 \leq s \leq t < \infty $, $ x \in E $, $ \Gamma \in \mathfrak B $, $ ( E , \mathfrak B ) $ being a measurable space), enabling one (under certain conditions on $ ( E , \mathfrak B ) $) to construct a Markov process for which the conditional probability $ {\mathsf P} _ {s,x} ( x _ {t} \in \Gamma ) $ is the same as $ P ( s , x ; t , \Gamma ) $. Conversely, for a Markov process its transition function $ P ( s , x ; t , \Gamma ) $, which by definition is equal to $ {\mathsf P} _ {s,x} ( x _ {t} \in \Gamma ) $, satisfies the Kolmogorov–Chapman equation, as follows immediately from general properties of conditional probabilities. This was pointed out by S. Chapman [C] and investigated by A.N. Kolmogorov in 1931 (see [K]).

References

[C] S. Chapman, "?", Proc. Roy. Soc. Ser. A , 119 (1928) pp. 34–54
[K] A. Kolmogoroff, "Ueber die analytischen Methoden in der Wahrscheinlichkeitsrechnung" Math. Ann. , 104 (1931) pp. 415–458
[GS] I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian)

Comments

In Western literature this equation is usually referred to as the Chapman–Kolmogorov equation.

See also (the editorial comments to) Einstein–Smoluchowski equation.

References

[L] P. Lévy, "Processus stochastiques et mouvement Brownien", Gauthier-Villars (1965)
[D] E.B. Dynkin, "Markov processes", 1, Springer (1965) pp. Sect. 5.26 (Translated from Russian)
[F] W. Feller, "An introduction to probability theory and its applications", 1, Wiley (1966) pp. Chapt. XV.13
How to Cite This Entry:
Kolmogorov-Chapman equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kolmogorov-Chapman_equation&oldid=22655
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article