the Ornstein-Uhlenbeck process
/ ORN-stine OO-len-bek /
Brownian motion wanders off to infinity — it has no home and never settles. But many real random quantities are tethered: a spring-loaded mass jiggling around its rest position, the velocity of a pollen grain slowed by friction, an interest rate or a commodity price that drifts but keeps being pulled back toward a long-run average. The Ornstein-Uhlenbeck process is the simplest model of such tethered randomness — Brownian noise plus a restoring pull toward a centre. It is the canonical 'mean-reverting' process: it jitters, but it always tends back home.
Its SDE is dX = -theta (X - mu) dt + sigma dB, with three knobs: mu is the centre it is pulled toward, theta greater than 0 is the strength of the pull (the reversion speed), and sigma is the size of the random kicks. Read the drift term -theta (X - mu): when X is above the centre mu it is negative (pull down), when X is below mu it is positive (pull up), and the further from home the stronger the pull — exactly a spring. The tug-of-war between the restoring drift and the random kicks reaches a balance, so unlike Brownian motion the OU process has a stationary distribution: started from far away it relaxes toward, and then fluctuates forever within, a fixed Normal distribution Normal(mu, sigma^2 / (2 theta)). It is in fact the ONLY process that is simultaneously Gaussian, Markov, and stationary.
The OU process is the noise model of choice whenever a quantity has a natural resting level. In physics it is the Langevin model of a particle's velocity under friction; in finance the Vasicek interest-rate model and many commodity and pairs-trading models are OU; in biology and engineering it models any damped, noise-driven system. It also has a clean exact solution (via an integrating-factor trick that survives in the Ito calculus), so you can write its mean, variance, and autocorrelation in closed form: the autocorrelation decays exponentially like exp(-theta times lag), meaning the process forgets its past at a rate set by theta. The honest caveat: the OU model assumes the pull is linear and the noise is constant Gaussian — real mean-reverting systems often revert nonlinearly or have noise that grows near boundaries, which OU does not capture.
Model a short-term interest rate pulled toward 3 percent with reversion speed theta = 2 per year and noise sigma = 1 percent. If the rate is currently 5 percent, its drift is -2 times (0.05 - 0.03) = -0.04 per year, pulling it down. Over the long run it settles into Normal(0.03, sigma^2/(2 theta)) = Normal(0.03, 0.0025), fluctuating around 3 percent.
A restoring drift pulls the process back home, so unlike Brownian motion it settles into a stationary Normal.
The OU process is the unique process that is Gaussian, Markov, and stationary all at once. Its mean reversion is LINEAR and its noise CONSTANT — real-world reverting systems often violate both.