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Difference between revisions of "Markov property"

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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) {{MR|0116388}} {{ZBL|0092.34304}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.L. Doob, "Stochastic processes" , Wiley (1953) {{MR|1570654}} {{MR|0058896}} {{ZBL|0053.26802}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E.B. Dynkin, "Markov processes" , '''1''' , Springer (1965) (Translated from Russian) {{MR|0193671}} {{ZBL|0132.37901}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> T.G. Kurtz, "Markov processes" , Wiley (1986) {{MR|0838085}} {{ZBL|0592.60049}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> W. Feller, "An introduction to probability theory and its applications" , '''1–2''' , Wiley (1966) {{MR|0210154}} {{ZBL|0138.10207}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) {{MR|0190953}} {{ZBL|0137.11602}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> M. Loève, "Probability theory" , '''II''' , Springer (1978) {{MR|0651017}} {{MR|0651018}} {{ZBL|0385.60001}} </TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) {{MR|0116388}} {{ZBL|0092.34304}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.L. Doob, "Stochastic processes" , Wiley (1953) {{MR|1570654}} {{MR|0058896}} {{ZBL|0053.26802}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E.B. Dynkin, "Markov processes" , '''1''' , Springer (1965) (Translated from Russian) {{MR|0193671}} {{ZBL|0132.37901}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> T.G. Kurtz, "Markov processes" , Wiley (1986) {{MR|0838085}} {{ZBL|0592.60049}} </TD></TR>
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<TR><TD valign="top">[a5]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its applications"]], '''1–2''' , Wiley (1966) </TD></TR>
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<TR><TD valign="top">[a6]</TD> <TD valign="top"> P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) {{MR|0190953}} {{ZBL|0137.11602}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> M. Loève, "Probability theory" , '''II''' , Springer (1978) {{MR|0651017}} {{MR|0651018}} {{ZBL|0385.60001}} </TD></TR></table>

Revision as of 18:24, 26 April 2012

for a real-valued stochastic process ,

2020 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

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


Comments

References

[a1] K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) MR0116388 Zbl 0092.34304
[a2] J.L. Doob, "Stochastic processes" , Wiley (1953) MR1570654 MR0058896 Zbl 0053.26802
[a3] E.B. Dynkin, "Markov processes" , 1 , Springer (1965) (Translated from Russian) MR0193671 Zbl 0132.37901
[a4] T.G. Kurtz, "Markov processes" , Wiley (1986) MR0838085 Zbl 0592.60049
[a5] W. Feller, "An introduction to probability theory and its applications", 1–2 , Wiley (1966)
[a6] P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) MR0190953 Zbl 0137.11602
[a7] 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=25529
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article