Markov property

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

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