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

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

References

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