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  "body": "\n## Summary\n\nThe Ornstein–Uhlenbeck process is a Brownian motion with a pull back towards a long-run mean.\nIt solves the [[Langevin equation|ZW2RBS9VTR]] $dX_t = \\theta(\\mu - X_t) \\, dt + \\sigma \\, dW_t$ with $\\theta \u003e 0$.\nIt is Gaussian, Markov and, from a suitable start, stationary, and it models velocities of particles, interest rates and mean-reverting quantities.\n\n## Explanations\n\n### L1 human\n\nThink of a particle tied to the point $\\mu$ by a spring, and kicked around at random.\nThe farther it strays, the harder the spring pulls it back, so it never wanders off for good.\nIn the long run its position is normally distributed around $\\mu$, with spread set by the balance of kicks $\\sigma$ and spring strength $\\theta$.\nUnlike Brownian motion, whose spread grows with time, this process settles into a steady random state.\n",
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