Markov Property

Definition

Markov property refers to the memoryless property of a Stochastic Process.

Consider a Probability Space $(\Omega ,\mathcal{F},P)$ with a Filtration $({\mathcal{F}}_s,s\in I)$, and a Measurable Space $(S,\mathcal{S})$. A Stochastic Process $X=\{X_t|\Omega \to S{\}}_{t\in I}$ is said to possess the Markov property if $\forall A\in \mathcal{S},\forall s,t\in I\text{s.t.}s<t,P(X_t\in A|{\mathcal{F}}_s)=P(X_t\in A|X_s)$ Here, ${\mathcal{F}}_s$ in the LHS represents all information up to time $s$, whereas $X_s$ conditions only on the single value at time $s$.

In the case where $S$ is a discrete set and $I=\mathbb{N}$, this can be reformulated as follows $\forall n\in \mathbb{N},P(X_{t+1}=x_{n+1}|X_n=x_n,\dots ,X_1=x_1)=P(X_{n+1}=x_{n+1}|X_n=x_n)$