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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] $(\\mathcal{F}_t)_{t \\ge 0}$.\nLet $(X_t)_{t \\ge 0}$ be a real [[adapted|YRNFE016FF]] process with càdlàg paths.\n$X$ is a *local martingale* if there is a sequence $(\\tau_n)_{n \\ge 1}$ of [[stopping times|WND87FBAKC]] such that:\n\n1. $\\tau_1 \\le \\tau_2 \\le \\dots$ almost surely.\n2. $\\tau_n \\to \\infty$ almost surely as $n \\to \\infty$.\n3. For each $n$, the [[stopped process|NG35MGYDZR]] $X^{\\tau_n} = (X_{t \\wedge \\tau_n})_{t \\ge 0}$ is a [[martingale|0CDVK9RQBH]].\n\nSuch a sequence $(\\tau_n)$ is called a localising sequence.\n\n## Remarks\n\nEvery càdlàg martingale $X$ is a local martingale: take $\\tau_n = n$, since a stopped martingale is a martingale by optional stopping.\nThe deterministic process $X_t = t$ is adapted and continuous but is not a local martingale.\nIndeed, clause 3 would give $E[t \\wedge \\tau_n] = 0$ for all $t$, so $\\tau_n = 0$ almost surely, contradicting clause 2.\nA local martingale need not be a martingale: if $B$ is a three-dimensional Brownian motion started at $x \\ne 0$, then $1/\\abs{B_t}$ is a local martingale but not a martingale.\nA non-negative local martingale is a [[supermartingale|KYW1PV0SX5]], by Fatou's lemma.\nSome authors require clause 3 only for $X^{\\tau_n} - X_0$, or for $X^{\\tau_n} \\mathbf{1}\\{\\tau_n \u003e 0\\}$, so that $X_0$ need not be integrable.\n",
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