Difference between revisions of "Markov property"
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[[Category:Markov processes]] | [[Category:Markov processes]] | ||
− | The property that for any set $ | + | The property that for any set $ t _ {1} < \dots < t _ {n+1} T and any Borel set B $, |
− | of times from T | ||
− | and any Borel set | ||
$$ \tag{* } | $$ \tag{* } | ||
− | {\mathsf P} \{ X ( t _ {n+} | + | {\mathsf P} \{ X ( t _ {n+1} ) \in B \mid X ( t _ {n} ) \dots X ( t _ {1} ) \} = |
− | \in B \mid X ( t _ {n} ) \dots X ( t _ {1} ) \} = | ||
$$ | $$ | ||
$$ | $$ | ||
= \ | = \ | ||
− | {\mathsf P} \{ X ( t _ {n+} | + | {\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+} | + | with probability 1, that is, the conditional probability distribution of $ X ( t _ {n+1} ) $ |
given X ( t _ {n} ) \dots X ( t _ {1} ) | given X ( t _ {n} ) \dots X ( t _ {1} ) | ||
− | coincides (almost certainly) with the conditional distribution of $ X ( t _ {n+} | + | coincides (almost certainly) with the conditional distribution of $ X ( t _ {n+1} ) $ |
given X ( t _ {n} ) . | given X ( t _ {n} ) . | ||
− | This can be interpreted as independence of the "future" $ X ( t _ {n+} | + | This can be interpreted as independence of the "future" $ X ( t _ {n+1} ) $ |
− | and the "past" $ ( X ( t _ {n-} | + | and the "past" $ ( X ( t _ {n-1} ) \dots X ( t _ {1} ) ) $ |
given the fixed "present" X ( t _ {n} ) . | given the fixed "present" X ( t _ {n} ) . | ||
Stochastic processes satisfying the property (*) are called Markov processes (cf. [[Markov process|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 \} | Stochastic processes satisfying the property (*) are called Markov processes (cf. [[Markov process|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 \} | ||
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|valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}} | |valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}} | ||
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|valign="top"|{{Ref|C}}|| K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) {{MR|0116388}} {{ZBL|0092.34304}} | |valign="top"|{{Ref|C}}|| K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) {{MR|0116388}} {{ZBL|0092.34304}} | ||
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Latest revision as of 19:53, 1 November 2023
for a real-valued stochastic process X ( t) ,
t \in T \subset \mathbf R
2020 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 |
[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 |
Markov property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_property&oldid=47775