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  "body": "\n## Summary\n\nThe Doléans-Dade exponential, or stochastic exponential, $\\mathcal{E}(X)$ of a [[semimartingale|4EJB6E2AZK]] $X$ is the solution of the equation $dZ = Z \\, dX$ with $Z_0 = 1$.\nIt plays the role of the exponential function in [[Itô calculus|JA95NH206E]].\nFor a continuous semimartingale with $X_0 = 0$ it equals $\\exp(X - \\tfrac12 [X])$.\nIt is the density process in the [[Girsanov theorem|Z76HD1BZ29]].\n\n## Explanations\n\n### L1 human\n\nIn ordinary calculus, the solution of $dz = z \\, dx$ with $z(0) = 1$ is $z = e^{x}$.\nFor a random driver such as Brownian motion, $e^{X}$ grows a little too fast, because Itô's lemma adds a drift.\nThe stochastic exponential removes this drift by the factor $e^{-[X]/2}$.\nFor $X = \\sigma B$ with a constant $\\sigma$, it is $\\exp(\\sigma B_t - \\tfrac12 \\sigma^2 t)$, a positive martingale with mean $1$.\n",
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