Markov property

Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space, let $T \subseteq [0, \infty)$, and let $(\mathcal{F}_t)_{t \in T}$ be a filtration: σ-algebras with $\mathcal{F}_s \subseteq \mathcal{F}_t \subseteq \mathcal{F}$ for $s \le t$. Let $(X_t)_{t \in T}$ be a stochastic process with values in $(S, \mathcal{S})$ such that each $X_t$ is $\mathcal{F}_t$-measurable. The process has the Markov property with respect to $(\mathcal{F}_t)$ if, for all $s \le t$ in $T$ and each $A \in \mathcal{S}$, $$P(X_t \in A \mid \mathcal{F}_s) = P(X_t \in A \mid X_s) \quad \text{almost surely}.$$

Remarks

Here $P(B \mid \mathcal{G}) = E[1_B \mid \mathcal{G}]$ is a conditional expectation. For a discrete-time process with a countable state space and the natural filtration, the property becomes $P(X_{n+1} = j \mid X_0, \dots, X_n) = P(X_{n+1} = j \mid X_n)$. This defines a Markov chain. The property says that the future is conditionally independent of the past, given the present. The strong Markov property is the same statement with a stopping time in place of the fixed time $s$.

Explanations

L1 human

A process with the Markov property forgets how it reached its present state. To predict its future, the present state is all the information that you need.