the Ornstein-Uhlenbeck process
/ ORN-stine OO-len-beck /
The Ornstein-Uhlenbeck process is the simplest non-trivial diffusion with a pull toward a fixed level: it is Brownian motion plus a linear restoring force. It is the canonical model of mean reversion — a noisy quantity that wanders but is continually dragged back toward an equilibrium. Physically it began as a model for the velocity of a particle undergoing friction and random kicks (the velocity of Brownian motion), and it now appears across finance (Vasicek interest rates), neuroscience (membrane potentials), and statistics (the stationary Gaussian Markov process).
It solves the linear SDE dX_t = -theta (X_t - m) dt + sigma dB_t, with mean-reversion rate theta > 0, long-run mean m, and noise level sigma. The drift -theta(X - m) points back toward m with strength proportional to the displacement; the diffusion sigma is constant (additive noise). Its great virtue is that it is exactly solvable: by the integrating-factor trick (apply Ito to e^(theta t) X_t) one gets X_t = m + (X_0 - m) e^(-theta t) + sigma integral_0^t e^(-theta(t-s)) dB_s. Because the noise enters linearly, X_t is Gaussian for every t: mean m + (X_0 - m)e^(-theta t), and variance (sigma^2/(2 theta))(1 - e^(-2 theta t)). As t -> infinity it forgets X_0 and relaxes to its invariant distribution N(m, sigma^2/(2 theta)). It is the unique stationary, Gaussian, Markovian process with continuous paths, and its stationary autocovariance decays exponentially: Cov(X_s, X_{s+t}) = (sigma^2/(2 theta)) e^(-theta t).
The OU process is the go-to example for almost every concept in this field: its generator is L f = -theta(x - m) f' + (sigma^2/2) f'', its Fokker-Planck steady state is the Gaussian above, and it is reversible with respect to that Gaussian (detailed balance holds). It is the diffusion underlying the OU semigroup and the hypercontractivity / log-Sobolev story for Gaussian measures. An honest contrast worth keeping: the OU process is mean-reverting and stationary, whereas plain Brownian motion (theta = 0) is not stationary and diffuses to infinity; and unlike geometric Brownian motion, the OU process can go negative, which is why it is unsuitable as a model for strictly positive quantities (prices, volatilities, rates that must stay positive) — there the CIR square-root process is used instead.
The Vasicek short-rate model dr_t = a(b - r_t) dt + sigma dB_t is an OU process for the interest rate, with mean reversion to b at speed a. It is analytically tractable (bond prices have closed form) but has the well-known flaw that r_t can become negative, since the OU stationary law is Gaussian and supported on all of R.
Vasicek interest rates: an OU process, exactly solvable but able to go negative.
The OU process is Gaussian and can take negative values, so it is wrong for strictly positive quantities — use CIR there. It is stationary only because theta > 0; at theta = 0 it degenerates to Brownian motion, which has no invariant probability law.