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Difference between revisions of "Kolmogorov-Chapman equation"

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(→‎References: Feller: internal link)
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====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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<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, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its  applications"]], '''1''', Wiley (1966) pp. Chapt. XV.13</TD></TR></table>

Revision as of 09:09, 4 May 2012

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

An equation of the form

that is, a condition imposed on the transition function (, , , being a measurable space), enabling one (under certain conditions on ) to construct a Markov process for which the conditional probability is the same as . Conversely, for a Markov process its transition function , which by definition is equal to , satisfies the Kolmogorov–Chapman equation, as follows immediately from general properties of conditional probabilities. This was pointed out by S. Chapman [1] and investigated by A.N. Kolmogorov in 1931 (see [2]).

References

[1] S. Chapman, Proc. Roy. Soc. Ser. A , 119 (1928) pp. 34–54
[2] A. [A.N. Kolmogorov] Kolmogoroff, "Ueber die analytischen Methoden in der Wahrscheinlichkeitsrechnung" Math. Ann. , 104 (1931) pp. 415–458
[3] I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] 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

[a1] P. Lévy, "Processus stochastiques et mouvement Brownien", Gauthier-Villars (1965)
[a2] E.B. Dynkin, "Markov processes", 1, Springer (1965) pp. Sect. 5.26 (Translated from Russian)
[a3] 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