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  "body": "\n## Statement\n\nLet $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t), P)$ be a filtered probability space, let $T \u003e 0$, and let $B$ be a [[Brownian motion|22735FGJV6]] for $(\\mathcal{F}_t)$ under $P$.\nLet $\\theta$ be a progressively measurable process with $\\int_0^T \\theta_s^2 \\, ds \u003c \\infty$ almost surely.\nLet $Z = \\mathcal{E}\\left(\\int_0^{\\cdot} \\theta_s \\, dB_s\\right)$ be the [[Doléans-Dade exponential|3D88622PBQ]]:\n$$Z_t = \\exp\\left(\\int_0^t \\theta_s \\, dB_s - \\frac12 \\int_0^t \\theta_s^2 \\, ds\\right), \\quad 0 \\le t \\le T.$$\nAssume $E_P[Z_T] = 1$, and let $Q$ be the [[probability measure|059TJN0NZ1]] on $\\mathcal{F}_T$ with $dQ = Z_T \\, dP$.\nThen the process\n$$\\tilde{B}_t = B_t - \\int_0^t \\theta_s \\, ds, \\quad 0 \\le t \\le T,$$\nis a Brownian motion for $(\\mathcal{F}_t)_{t \\le T}$ under $Q$.\n\n## Remarks\n\nThe [[Itô integral|JA95NH206E]] in the exponent makes $Z$ a positive [[local martingale|XC2YPC2BSW]] under $P$, and the hypothesis $E_P[Z_T] = 1$ makes it a true martingale on $[0, T]$.\nBy Itô's product rule, $Z \\tilde{B}$ is a local martingale under $P$, so $\\tilde{B}$ is a continuous local martingale under $Q$.\nIts quadratic variation is still $t$, so [[Lévy's characterization|8CKDVESXT6]] shows that it is a Brownian motion under $Q$.\nNovikov's condition $E_P\\left[\\exp\\left(\\tfrac12 \\int_0^T \\theta_s^2 \\, ds\\right)\\right] \u003c \\infty$ implies $E_P[Z_T] = 1$.\nExample: for a constant $\\theta = \\mu$, the density is $Z_T = \\exp(\\mu B_T - \\tfrac12 \\mu^2 T)$, and under $Q$ the process $B$ is a Brownian motion with drift $\\mu$.\nWithout $E_P[Z_T] = 1$, the measure $Q$ has total mass less than $1$ and is not a probability measure.\n\n## Explanations\n\n### L1 human\n\nChanging the probabilities of the paths in a suitable way can turn a Brownian motion with drift into a Brownian motion without drift.\nThe theorem gives the exact weight for each path.\nThe new and old probability measures agree on which events are impossible.\n\n### L2 human\n\nGirsanov's theorem is the standard way to remove or add a drift in a stochastic differential equation.\nIn mathematical finance, it gives the risk-neutral measure under which discounted prices are martingales.\nIt also yields weak solutions of equations $dX = b(X) \\, dt + dB$ with a bounded measurable drift $b$, and likelihood ratios for statistics of diffusions.\nThe theorem extends to general continuous local martingales in place of $B$.\n",
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