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  "body": "\n## Definition\n\nLet $(W_t)_{t \\ge 0}$ be a [[Wiener process|552AWX3QXN]], let $X_0$ be a real random variable independent of $W$, and let $\\theta \u003e 0$, $\\sigma \u003e 0$ and $\\mu \\in \\R$.\nThe *Ornstein–Uhlenbeck process* with these parameters is the solution $(X_t)_{t \\ge 0}$ of the stochastic differential equation\n$$dX_t = \\theta (\\mu - X_t) \\, dt + \\sigma \\, dW_t$$\nwith initial value $X_0$.\nExplicitly,\n$$X_t = \\mu + (X_0 - \\mu) e^{-\\theta t} + \\sigma \\int_0^t e^{-\\theta (t - s)} \\, dW_s.$$\n\n## Remarks\n\nThe explicit formula solves the equation: Itô's lemma applied to $e^{\\theta t} X_t$ gives $d(e^{\\theta t} X_t) = \\theta \\mu e^{\\theta t} \\, dt + \\sigma e^{\\theta t} \\, dW_t$, which integrates to the formula.\nIf $X_0$ is constant, $X_t$ is [[normal|05AQE0QR0D]] with mean $\\mu + (X_0 - \\mu) e^{-\\theta t}$ and variance $\\frac{\\sigma^2}{2\\theta}(1 - e^{-2\\theta t})$.\nIf $X_0$ is normal with mean $\\mu$ and variance $\\sigma^2 / (2\\theta)$, the process is [[stationary|DT0QXJE705]], with covariance $\\frac{\\sigma^2}{2\\theta} e^{-\\theta \\abs{t - s}}$.\nFor $\\theta = 0$, which the definition excludes, the equation gives $X_0 + \\sigma W_t$, a Brownian motion with no mean reversion and no stationary law.\nUp to linear changes of space and time variables, it is the only non-trivial process that is stationary, Gaussian and Markov.\n",
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