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Markov property

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for a real-valued stochastic process ,

2010 Mathematics Subject Classification: Primary: 60Jxx [MSN][ZBL]

The property that for any set of times from and any Borel set ,

(*)

with probability 1, that is, the conditional probability distribution of given coincides (almost certainly) with the conditional distribution of given . This can be interpreted as independence of the "future" and the "past" given the fixed "present" . Stochastic processes satisfying the property (*) are called Markov processes (cf. Markov process). The Markov property has (under certain additional assumptions) a stronger version, known as the "strong Markov property" . In discrete time the strong Markov property, which is always true for (Markov) sequences satisfying (*), means that for each stopping time (relative to the family of -algebras , ), with probability one

References

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

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References

[C] K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) MR0116388 Zbl 0092.34304
[Do] J.L. Doob, "Stochastic processes" , Wiley (1953) MR1570654 MR0058896 Zbl 0053.26802
[Dy] E.B. Dynkin, "Markov processes" , 1 , Springer (1965) (Translated from Russian) MR0193671 Zbl 0132.37901
[K] T.G. Kurtz, "Markov processes" , Wiley (1986) MR0838085 Zbl 0592.60049
[F] W. Feller, "An introduction to probability theory and its applications", 1–2 , Wiley (1966)
[Le] P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) MR0190953 Zbl 0137.11602
[Lo] M. Loève, "Probability theory" , II , Springer (1978) MR0651017 MR0651018 Zbl 0385.60001
How to Cite This Entry:
Markov property. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Markov_property&oldid=26609
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article