1 Markov Property
In probability theory the term Markov property refers to the memoryless property of a stochastic process, which means that its future evolution is independent of its history.
Let \((\Omega, \mathcal{F}, \mathbb{P})\) be a [[knowledge-mathematics-analysis-probability-theory-probability-spaces|probability space]] with a filtration \((\mathcal{F}_{s},~s \in I)\) for some [[total-order|totally ordered]] index set \(I\); and let \((S,\mathcal{S})\) be a measurable space. A \((S,\mathcal{S})\)-valued stochastic process \(X=\{ X_{t}:\Omega \to S \}_{t \in I}\) adapted to the filtration is said to possess the Markov property if, for each \(A \in \mathcal{S}\) and each \(s,t \in I\) with \(s<t\), \[\mathbb{P}(X_{t} \in S|\mathcal{F}_{s})=\mathbb{P}(X_{t} \in A|X_{s}).\]In the case where \(S\) is a discrete set with the discrete [[sigma-algebra|\(\sigma\)-algebra]] and \(I=\mathbb{N}\), this is reformulated as \[\mathbb{P}(X_{n+1}=x_{n+1}|X_{n}=x_{n}, \dots, X_{1}=x_1)=\mathbb{P}(X_{n+1}=x_{n+1}|X_{n}=x_{n}),\] for all \(n \in \mathbb{N}\).
An alternative property can be formulated as \[ \mathbb{E}[f(X_{t})|\mathcal{F}_{s}]=\mathbb{E}[f(X_{t})|\sigma(X_{s})], \] for all \(t\geq s\geq 0\) and \(f:S\to \mathbb{R}\).
1.1 Strong Markov Property
The strong Markov property is based on the same concept with the key difference that the time, say \(\tau\), that the present refers to is a stopping time.
Suppose that \(X=(X_{t},~t\geq 0)\) is a stochastic process on a [[knowledge-mathematics-analysis-probability-theory-probability-spaces|probability space]] \((\Omega, \mathcal{F}, \mathbb{P})\) with natural filtration \(\{ \mathcal{F}_{t} \}_{t\geq 0}\). Then, for any stopping time \(\tau\) on \(\Omega\) we can define \[ \mathcal{F}_{\tau}=\{ A\in\mathcal{F}:\forall t\geq 0,\{ \tau \leq t \}\cap A\in \mathcal{F}_{t} \} \] to be the information up to \(\tau\). Then \(X\) is said to have the strong Markov property if, for each stopping time \(\tau\), conditional on the event \(\{ \tau<\infty \}\) we have that for each \(t\geq 0\), \(X_{\tau+t}\) is independent of \(\mathcal{F}_{\tau}\) given \(X_{\tau}\).