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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]], where $T \\subseteq [0, \\infty)$, for example $T = \\N$ or $T = [0, \\infty)$.\nA *stopping time* with respect to $(\\mathcal{F}_t)$ is a map $\\tau \\colon \\Omega \\to T \\cup \\set{\\infty}$ such that\n$$\\set{\\omega \\in \\Omega : \\tau(\\omega) \\le t} \\in \\mathcal{F}_t \\quad \\text{for every } t \\in T.$$\n\n## Remarks\n\nIn discrete time, $T = \\N$, the condition is equivalent to $\\set{\\tau = n} \\in \\mathcal{F}_n$ for every $n \\in \\N$, since $\\set{\\tau = n} = \\set{\\tau \\le n} \\setminus \\set{\\tau \\le n - 1}$ and $\\set{\\tau \\le n} = \\bigcup_{k \\le n} \\set{\\tau = k}$.\nThe value $\\infty$ means that the event never happens.\n\nExample: let $(S_n)_{n \\in \\N}$ be a [[random walk|04VZE3FZXF]] with natural filtration $\\mathcal{F}_n = \\sigma(S_0, \\dots, S_n)$, let $a \\in \\R$, and let $\\tau = \\min\\set{n : S_n \\ge a}$, with $\\min \\emptyset = \\infty$.\nThen $\\set{\\tau \\le n} = \\bigcup_{k \\le n} \\set{S_k \\ge a} \\in \\mathcal{F}_n$, so $\\tau$ is a stopping time.\n\nNon-example: with the same walk started at $S_0 = 0$ and with steps $\\pm 1$, let $N \\ge 2$ and $\\sigma = \\max\\set{n \\le N : S_n = 0}$, the last visit to $0$ before time $N$.\nThe event $\\set{\\sigma \\le 0}$ is the event that $S_n \\ne 0$ for $1 \\le n \\le N$, which depends on steps after time $0$ and is not in $\\mathcal{F}_0 = \\set{\\emptyset, \\Omega}$.\nSo $\\sigma$ is not a stopping time.\n\nConstant times are stopping times, and the minimum and the maximum of two stopping times are stopping times.\nThe *$\\sigma$-algebra of the past* at $\\tau$ is $\\mathcal{F}_\\tau = \\set{A \\in \\mathcal{F} : A \\cap \\set{\\tau \\le t} \\in \\mathcal{F}_t \\text{ for all } t \\in T}$.\n",
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