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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_{t \\ge 0}, P)$ be a [[probability space|0NFPJVJJHX]] with a [[filtration|HNK1Y62G7V]] that satisfies the usual conditions.\nLet $W = (W_t)_{t \\ge 0}$ be an $m$-dimensional [[Wiener process|22735FGJV6]] with respect to this filtration.\nLet $T \u003e 0$, and let $b \\colon [0, T] \\times \\R^n \\to \\R^n$ and $\\sigma \\colon [0, T] \\times \\R^n \\to \\R^{n \\times m}$ be Borel measurable.\nLet $\\xi$ be an $\\R^n$-valued $\\mathcal{F}_0$-measurable random variable.\nThe *stochastic differential equation* with drift $b$, diffusion coefficient $\\sigma$ and initial value $\\xi$ is\n$$dX_t = b(t, X_t) \\, dt + \\sigma(t, X_t) \\, dW_t, \\qquad X_0 = \\xi.$$\nA *strong solution* is a continuous [[adapted process|YRNFE016FF]] $X$ with values in $\\R^n$ with two properties.\nAlmost surely, $\\int_0^T \\left( \\abs{b(s, X_s)} + \\abs{\\sigma(s, X_s)}^2 \\right) ds \u003c \\infty$.\nAlmost surely,\n$$X_t = \\xi + \\int_0^t b(s, X_s) \\, ds + \\int_0^t \\sigma(s, X_s) \\, dW_s \\quad \\text{for all } t \\in [0, T],$$\nwhere the last integral is the Itô integral of [[Itô calculus|JA95NH206E]].\n\n## Remarks\n\nAssume that $b$ and $\\sigma$ are [[Lipschitz|0AW1E6YTJY]] in $x$ uniformly in $t$ and grow at most linearly in $x$, and that $\\xi$ has a finite second moment.\nThen a strong solution exists and is unique up to indistinguishability.\nExample: for $n = m = 1$, $b(t, x) = \\mu x$ and $\\sigma(t, x) = s x$ with constants $\\mu, s$, the process $X_t = X_0 \\exp\\left( (\\mu - s^2/2) t + s W_t \\right)$ is a solution; [[Itô's lemma|2J1RVQB0CM]] gives $dX_t = \\mu X_t \\, dt + s X_t \\, dW_t$.\nThis is [[geometric Brownian motion|D48DR46B97]].\nNon-example: $Y_t = e^{W_t}$ does not solve $dY_t = Y_t \\, dW_t$, $Y_0 = 1$, because Itô's lemma gives $dY_t = Y_t \\, dW_t + \\frac{1}{2} Y_t \\, dt$, with an extra drift term.\nA *weak solution* allows the probability space and the Wiener process to be constructed together with $X$.\n",
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