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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space with a [[filtration|HNK1Y62G7V]] $(\\mathcal{F}_t)_{t \\ge 0}$ that is right-continuous and complete.\nA function $a \\colon [0, \\infty) \\to \\R$ has finite variation if $\\sup \\sum_{k} \\abs{a(t_k) - a(t_{k-1})} \u003c \\infty$ on each interval $[0, t]$, the supremum being over finite partitions $0 = t_0 \u003c \\dots \u003c t_m = t$.\nA real [[adapted|YRNFE016FF]] process $(X_t)_{t \\ge 0}$ with càdlàg paths is a *semimartingale* if it can be written as\n$$X_t = X_0 + M_t + A_t, \\qquad t \\ge 0,$$\nwhere:\n\n1. $(M_t)$ is a [[local martingale|ZWJ9G89W3T]] with $M_0 = 0$.\n2. $(A_t)$ is an adapted càdlàg process with $A_0 = 0$ whose paths have finite variation almost surely.\n\n## Remarks\n\nBrownian motion with drift, $X_t = W_t + \\mu t$ with $W$ a Wiener process and $\\mu \\in \\R$, is a semimartingale: $M = W$ is a martingale and $A_t = \\mu t$ has finite variation.\n[[Fractional Brownian motion|VC6R7XSKRQ]] with Hurst index $H \\ne 1/2$ is not a semimartingale for its own filtration, so it admits no such decomposition.\nThe decomposition is not unique in general; it is unique when $A$ is required to be predictable and such a decomposition exists, as for continuous semimartingales.\nBy the Bichteler–Dellacherie theorem, semimartingales are exactly the good integrators: the adapted càdlàg processes for which the stochastic integral of simple predictable processes extends continuously.\nEvery semimartingale has a [[quadratic variation|ZBYC3MFW89]], and [[Itô's lemma|2J1RVQB0CM]] holds for $C^2$ functions of semimartingales.\n",
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