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  "body": "\n## Definition\n\nLet $(W_t)_{t \\ge 0}$ be a [[Wiener process|552AWX3QXN]], and let $\\mu \\in \\R$, $\\sigma \u003e 0$ and $s_0 \u003e 0$.\nThe *geometric Brownian motion* with drift $\\mu$, volatility $\\sigma$ and initial value $s_0$ is the process $(S_t)_{t \\ge 0}$ given by\n$$S_t = s_0 \\exp\\Bigl(\\bigl(\\mu - \\tfrac{\\sigma^2}{2}\\bigr) t + \\sigma W_t\\Bigr).$$\n\n## Remarks\n\nBy [[Itô's lemma|2J1RVQB0CM]], $S$ is the unique strong solution of $dS_t = \\mu S_t \\, dt + \\sigma S_t \\, dW_t$ with $S_0 = s_0$.\nCheck: with $f(t, x) = s_0 \\exp((\\mu - \\sigma^2/2) t + \\sigma x)$, Itô's lemma gives $dS_t = (\\mu - \\sigma^2/2) S_t \\, dt + \\sigma S_t \\, dW_t + \\tfrac{1}{2} \\sigma^2 S_t \\, dt$, which simplifies to the equation above.\nThe arithmetic process $s_0 + \\mu t + \\sigma W_t$ is not a geometric Brownian motion: it takes negative values with positive probability, while $S_t \u003e 0$ always.\nFor each $t$, $\\log S_t$ is normal with mean $\\log s_0 + (\\mu - \\sigma^2/2) t$ and variance $\\sigma^2 t$, so $S_t$ is [[log-normal|4W40NHWPQT]].\nIts mean is $E[S_t] = s_0 e^{\\mu t}$, and for $\\mu = 0$ the process is a martingale.\n",
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