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  "body": "\n## Definition\n\nLet $T \u003e 0$, let $x_0 \\in \\R$, and let $a, b \\colon [0, T] \\times \\R \\to \\R$ be functions.\nConsider the stochastic differential equation $dX_t = a(t, X_t) \\, dt + b(t, X_t) \\, dB_t$ with $X_0 = x_0$, where $B$ is a [[Brownian motion|22735FGJV6]].\nLet $N \\ge 1$, let $\\Delta t = T / N$, and let $t_n = n \\, \\Delta t$ for $0 \\le n \\le N$.\nThe *Euler–Maruyama approximation* is the sequence $(Y_n)_{0 \\le n \\le N}$ with $Y_0 = x_0$ and\n$$Y_{n+1} = Y_n + a(t_n, Y_n) \\, \\Delta t + b(t_n, Y_n) \\, \\Delta B_n, \\quad \\Delta B_n = B_{t_{n+1}} - B_{t_n}.$$\n\n## Remarks\n\nThe increments $\\Delta B_n$ are independent and [[normal|05AQE0QR0D]] with mean $0$ and variance $\\Delta t$.\nIn a simulation one takes $\\Delta B_n = \\sqrt{\\Delta t} \\, Z_n$ with independent standard normal $Z_n$.\nExample: for [[geometric Brownian motion|D48DR46B97]], $a(t, x) = \\mu x$ and $b(t, x) = \\sigma x$, so $Y_{n+1} = Y_n (1 + \\mu \\Delta t + \\sigma \\Delta B_n)$.\nNon-example: the update $Y_{n+1} = Y_n + a(t_n, Y_n) \\Delta t + b(t_n, Y_n) \\Delta t \\, Z_n$ uses noise of size $\\Delta t$ instead of $\\sqrt{\\Delta t}$, so its noise vanishes in the limit.\nIf $b = 0$, the scheme is the [[Euler method|0JARTXX4XP]] for the ordinary equation $x' = a(t, x)$.\nAssume that $a$ and $b$ are Lipschitz in $x$ uniformly in $t$, grow at most linearly, and are Hölder continuous of exponent $1/2$ in $t$.\nThen $E\\abs{X_T - Y_N} \\le C \\sqrt{\\Delta t}$ for a constant $C$.\n",
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