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Markov property

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for a real-valued stochastic process $ X ( t) $, $ t \in T \subset \mathbf R $

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

The property that for any set $ t _ {1} < \dots < t _ {n+} 1 $ of times from $ T $ and any Borel set $ B $,

$$ \tag{* } {\mathsf P} \{ X ( t _ {n+} 1 ) \in B \mid X ( t _ {n} ) \dots X ( t _ {1} ) \} = $$

$$ = \ {\mathsf P} \{ X ( t _ {n+} 1 ) \in B \mid X ( t _ {n} ) \} $$

with probability 1, that is, the conditional probability distribution of $ X ( t _ {n+} 1 ) $ given $ X ( t _ {n} ) \dots X ( t _ {1} ) $ coincides (almost certainly) with the conditional distribution of $ X ( t _ {n+} 1 ) $ given $ X ( t _ {n} ) $. This can be interpreted as independence of the "future" $ X ( t _ {n+} 1 ) $ and the "past" $ ( X ( t _ {n-} 1 ) \dots X ( t _ {1} ) ) $ given the fixed "present" $ X ( t _ {n} ) $. 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 $ T = \{ 1 , 2 ,\dots \} $ the strong Markov property, which is always true for (Markov) sequences satisfying (*), means that for each stopping time $ \tau $( relative to the family of $ \sigma $- algebras $ ( F _ {n} , n \geq 1) $, $ F _ {n} = \sigma \{ \omega : {X ( 1) \dots X ( n) } \} $), with probability one

$$ {\mathsf P} \{ X ( \tau + 1 ) \in B \mid X ( \tau ) \dots X ( 1) \} = $$

$$ = \ {\mathsf P} \{ X ( \tau + 1 ) \in B \mid X ( \tau ) \} . $$

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://encyclopediaofmath.org/index.php?title=Markov_property&oldid=47775
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article