Editing Markov property (section)

==Definition==

Let <math>(\Omega,\mathcal{F},P)</math> be a [[probability space]] with a [[Filtration (probability theory)|filtration]] <math>(\mathcal{F}_s,\ s \in I)</math>, for some ([[totally ordered]]) index set <math>I</math>; and let <math>(S,\mathcal{S})</math> be a [[measurable space]]. A <math>(S,\mathcal{S})</math>-valued stochastic process <math>X=\{X_t:\Omega \to S\}_{t\in I}</math> [[Adapted process|adapted to the filtration]] is said to possess the ''Markov property'' if, for each <math>A \in \mathcal{S}</math>  and each <math>s,t\in I</math> with <math>s<t</math>,

:<math>P(X_t \in A \mid \mathcal{F}_s) = P(X_t \in A\mid X_s).</math><ref>[[Rick Durrett|Durrett, Rick]]. ''Probability: Theory and Examples''. Fourth Edition. [[Cambridge University Press]], 2010.</ref>

In the case where <math>S</math> is a discrete set with the [[Sigma-algebra#Simple set-based examples|discrete sigma algebra]] and <math>I = \mathbb{N}</math>, this can be reformulated as follows:

:<math>P(X_{n+1}=x_{n+1}\mid X_n=x_n, \dots, X_1=x_1)=P(X_{n+1}=x_{n+1}\mid X_n=x_n) \text{ for all } n \in \mathbb{N}.</math>