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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]], and let $T$ be a totally ordered index set, such as $\\N$, $\\set{0, 1, \\dots, N}$ or $[0, \\infty)$.\nA *filtration* is a family $(\\mathcal{F}_t)_{t \\in T}$ of [[sigma-algebras|0SV8K0TDGY]] on $\\Omega$ such that:\n\n1. $\\mathcal{F}_t \\subseteq \\mathcal{F}$ for every $t \\in T$;\n2. $\\mathcal{F}_s \\subseteq \\mathcal{F}_t$ whenever $s, t \\in T$ and $s \\le t$.\n\nThe quadruple $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_{t \\in T}, P)$ is a *filtered probability space*.\n\n## Remarks\n\nExample: for a [[stochastic process|0QH5ENKZM1]] $(X_t)_{t \\in T}$, the *natural filtration* is $\\mathcal{F}_t^X = \\sigma(X_s : s \\le t)$, the smallest $\\sigma$-algebra for which all $X_s$ with $s \\le t$ are measurable.\nIf $s \\le t$, every generator of $\\mathcal{F}_s^X$ is a generator of $\\mathcal{F}_t^X$, so clause 2 holds.\n\nNon-example: let $X$ be a random variable that is not almost surely constant, so that $\\sigma(X) \\neq \\set{\\emptyset, \\Omega}$.\nOn $T = \\set{0, 1}$, the family $\\mathcal{F}_0 = \\sigma(X)$, $\\mathcal{F}_1 = \\set{\\emptyset, \\Omega}$ fails clause 2, since $\\mathcal{F}_0 \\not\\subseteq \\mathcal{F}_1$: information would be forgotten.\n\nIn continuous time one often assumes the *usual conditions*: $\\mathcal{F}_0$ contains all $P$-null sets of the completion of $\\mathcal{F}$, and the filtration is right-continuous, that is $\\mathcal{F}_t = \\bigcap_{s \u003e t} \\mathcal{F}_s$ for every $t$.\n",
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