Ornstein-Uhlenbeck process
Quantitative Finance glossary · Market Microstructure & Statistical Arbitrage
Definition
Mean-reverting Gaussian process: dX_t = κ(θ - X_t),dt + σ,dW_t, with closed-form solution X_t = θ + (X_0 - θ)e^-κ t + σint_0^t e^-κ(t-s)dW_s. Stationary distribution mathcalN(θ, σ^2/(2κ)); half-life t_1/2 = ln 2 / κ. Foundation of the Vasicek short-rate model, equity-pairs-trade spread dynamics, and the variance process in normal stochastic-vol models.
Ornstein-Uhlenbeck process in plain English
Picture a rubber band tied to a fixed peg: pull the ball away and it gets tugged back toward the peg, but random jostling keeps knocking it around so it never sits perfectly still. That's an Ornstein-Uhlenbeck process — a quantity that randomly wanders but is always pulled back toward a long-run average θ, with κ setting how strongly it snaps back and σ how wild the random jostling is. Because it always returns home, it's perfect for modeling things that don't drift off forever — like interest rates, or the price *gap* between two similar stocks that a "pairs trader" bets will close. A handy number is its half-life t_1/2 = ln 2 / κ: how long, on average, it takes to erase half of any push away from center.
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