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''for a real-valued [[Stochastic process|stochastic process]]  $  X ( t) $, 
$  t \in T \subset  \mathbf R $''

{{MSC|60Jxx}}

[[Category:Markov processes]]

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|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====
{|
|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|C}}|| K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) {{MR|0116388}} {{ZBL|0092.34304}}
|-
|valign="top"|{{Ref|Do}}|| J.L. Doob, "Stochastic processes" , Wiley (1953) {{MR|1570654}} {{MR|0058896}} {{ZBL|0053.26802}}
|-
|valign="top"|{{Ref|Dy}}|| E.B. Dynkin, "Markov processes" , '''1''' , Springer (1965) (Translated from Russian) {{MR|0193671}} {{ZBL|0132.37901}}
|-
|valign="top"|{{Ref|K}}|| T.G. Kurtz, "Markov processes" , Wiley (1986) {{MR|0838085}} {{ZBL|0592.60049}}
|-
|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1–2''' , Wiley (1966)
|-
|valign="top"|{{Ref|Le}}|| P. Lévy, "Processus stochastiques et mouvement Brownien" , Gauthier-Villars (1965) {{MR|0190953}} {{ZBL|0137.11602}}
|-
|valign="top"|{{Ref|Lo}}|| M. Loève, "Probability theory" , '''II''' , Springer (1978) {{MR|0651017}} {{MR|0651018}} {{ZBL|0385.60001}}
|}