Difference between revisions of "Kolmogorov-Chapman equation"
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| + | $#C+1 = 11 : ~/encyclopedia/old_files/data/K055/K.0505680 Kolmogorov\ANDChapman equation | ||
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| + | {{MSC|60J35}} | ||
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| + | [[Category:Markov processes]] | ||
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An equation of the form | 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|transition function]] | + | that is, a condition imposed on the [[Transition function|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|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 {{Cite|C}} and investigated by A.N. Kolmogorov in 1931 (see {{Cite|K}}). | ||
====References==== | ====References==== | ||
| − | + | {| | |
| − | + | |valign="top"|{{Ref|C}}|| S. Chapman, "?", ''Proc. Roy. Soc. Ser. A'' , '''119''' (1928) pp. 34–54 | |
| − | + | |- | |
| + | |valign="top"|{{Ref|K}}|| A. Kolmogoroff, "Ueber die analytischen Methoden in der Wahrscheinlichkeitsrechnung" ''Math. Ann.'' , '''104''' (1931) pp. 415–458 | ||
| + | |- | ||
| + | |valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russian) | ||
| + | |} | ||
====Comments==== | ====Comments==== | ||
| Line 16: | Line 52: | ||
====References==== | ====References==== | ||
| − | + | {| | |
| + | |valign="top"|{{Ref|L}}|| P. Lévy, "Processus stochastiques et mouvement Brownien", Gauthier-Villars (1965) | ||
| + | |- | ||
| + | |valign="top"|{{Ref|D}}|| E.B. Dynkin, "Markov processes", '''1''', Springer (1965) pp. Sect. 5.26 (Translated from Russian) | ||
| + | |- | ||
| + | |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 | ||
| + | |} | ||
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=13649
Kolmogorov-Chapman equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kolmogorov-Chapman_equation&oldid=13649
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article