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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ 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 $s \\le t$.\nLet $(X_t)_{t \\in T}$ be a [[stochastic process|0QH5ENKZM1]] with values in $(S, \\mathcal{S})$ such that each $X_t$ is $\\mathcal{F}_t$-measurable.\nThe process has the Markov property with respect to $(\\mathcal{F}_t)$ if, for all $s \\le t$ in $T$ and each $A \\in \\mathcal{S}$,\n$$P(X_t \\in A \\mid \\mathcal{F}_s) = P(X_t \\in A \\mid X_s) \\quad \\text{almost surely}.$$\n\n## Remarks\n\nHere $P(B \\mid \\mathcal{G}) = E[1_B \\mid \\mathcal{G}]$ is a [[conditional expectation|0YAMJRK04E]].\nFor 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.\nThe property says that the future is [[conditionally independent|0ZJ87ZZ8PK]] of the past, given the present.\nThe strong Markov property is the same statement with a stopping time in place of the fixed time $s$.\n\n## Explanations\n\n### L1 human\n\nA process with the Markov property forgets how it reached its present state.\nTo predict its future, the present state is all the information that you need.\n",
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