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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_{t \\in T}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] indexed by a totally ordered set $T$.\nLet $(S, \\mathcal{S})$ be a [[measurable|YYNCJSR0PJ]] space.\nA [[stochastic process|0QH5ENKZM1]] $X = (X_t)_{t \\in T}$ with values in $S$ is *adapted* to $(\\mathcal{F}_t)$ if the following holds.\nFor every $t \\in T$, the map $X_t \\colon \\Omega \\to S$ is measurable with respect to $\\mathcal{F}_t$ and $\\mathcal{S}$.\n\n## Remarks\n\nEvery process is adapted to its natural filtration $\\mathcal{F}_t^X = \\sigma(X_s : s \\le t)$, and this is the smallest filtration to which it is adapted.\n\nExample: let $Y_1, Y_2, \\dots$ be random variables with natural filtration $\\mathcal{F}_n = \\sigma(Y_1, \\dots, Y_n)$.\nThe partial sums $S_n = Y_1 + \\cdots + Y_n$ form an adapted process, since $S_n$ is a measurable function of $Y_1, \\dots, Y_n$.\n\nNon-example: let the $Y_k$ be independent fair signs, equal to $1$ or $-1$, with the same filtration, and let $X_n = Y_{n+1}$.\nThen $X_n$ is independent of $\\mathcal{F}_n$ and not almost surely constant, so it is not $\\mathcal{F}_n$-measurable; the process $X$ looks one step into the future.\n\nIn continuous time a stronger notion is often needed: $X$ is *progressively measurable* if, for every $t$, the map $(s, \\omega) \\mapsto X_s(\\omega)$ on $[0, t] \\times \\Omega$ is measurable with respect to $\\mathcal{B}([0, t]) \\otimes \\mathcal{F}_t$.\nAn adapted process with right-continuous or left-continuous paths is progressively measurable.\n",
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