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  "body": "\n## Definition\n\nLet $X$ be a real [[semimartingale|4EJB6E2AZK]] on a filtered probability space.\nThe *Doléans-Dade exponential* of $X$ is the unique semimartingale $Z = \\mathcal{E}(X)$ such that, for all $t \\ge 0$,\n$$Z_t = 1 + \\int_0^t Z_{s-} \\, dX_s.$$\nHere the integral is the stochastic integral of [[Itô calculus|JA95NH206E]].\nThe process $Z_{s-}$ is the left limit of $Z$ at $s$.\n\n## Remarks\n\nIf $X$ is continuous, then $\\mathcal{E}(X)_t = \\exp\\left(X_t - X_0 - \\tfrac12 [X]_t\\right)$, where $[X]$ is the [[quadratic variation|X4DHMZD5JA]] of $X$.\nIn general, $\\mathcal{E}(X)_t = \\exp\\left(X_t - X_0 - \\tfrac12 [X]^c_t\\right) \\prod_{0 \u003c s \\le t} (1 + \\Delta X_s) e^{-\\Delta X_s}$, where $[X]^c$ is the continuous part of $[X]$ and $\\Delta X_s = X_s - X_{s-}$.\nExample: let $B$ be a [[Brownian motion|22735FGJV6]] and $X = \\sigma B$ with a constant $\\sigma$.\nThen $Z_t = \\exp(\\sigma B_t - \\tfrac12 \\sigma^2 t)$, and [[Itô's lemma|2J1RVQB0CM]] gives $dZ_t = \\sigma Z_t \\, dB_t = Z_t \\, dX_t$.\nNon-example: $W_t = \\exp(\\sigma B_t)$ fails the equation, since Itô's lemma gives $dW_t = W_t \\, dX_t + \\tfrac12 \\sigma^2 W_t \\, dt$.\nIf $X$ is a continuous [[local martingale|XC2YPC2BSW]], then $\\mathcal{E}(X)$ is a positive local martingale and hence a supermartingale.\nIt is a true martingale under Novikov's condition $E\\left[\\exp\\left(\\tfrac12 [X]_\\infty\\right)\\right] \u003c \\infty$.\n",
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