Markov property
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 |
Comments
References
[C] | K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) MR0116388 Zbl 0092.34304 |
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[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 |
Markov property. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Markov_property&oldid=26609