The Ornstein-Uhlenbeck Process

Core Idea

The Ornstein-Uhlenbeck (OU) process solves dX = -θX dt + σ dW, where θ > 0 controls the rate of mean reversion. Its explicit solution X(t) = X(0)e^{-θt} + σ∫₀ᵗ e^{-θ(t-s)} dW(s) is Gaussian with mean X(0)e^{-θt} and variance (σ²/2θ)(1 - e^{-2θt}). As t → ∞, the process converges to a stationary Gaussian distribution N(0, σ²/2θ). The OU process is the prototypical mean-reverting diffusion and the only stationary Gaussian Markov process.

Explainer

The Ornstein-Uhlenbeck process is the simplest non-trivial SDE with an explicit solution and a non-degenerate stationary distribution. It satisfies dX = -θX dt + σ dW, where the drift -θX acts as a restoring force pulling the process toward zero. When X is positive, the drift is negative (pushing down); when X is negative, the drift is positive (pushing up). This is mean reversion — the continuous-time analogue of a discrete-time AR(1) process with coefficient e^{-θ}.

The solution technique uses an integrating factor, paralleling the method for linear ODEs. Define Y(t) = X(t)e^{θt}. By Itô's formula, dY = e^{θt}(dX + θX dt) = e^{θt}σ dW. This is a pure Itô integral with no drift, so Y(t) = X(0) + σ∫₀ᵗ e^{θs} dW(s). Multiplying by e^{-θt} gives the explicit solution: X(t) = X(0)e^{-θt} + σ∫₀ᵗ e^{-θ(t-s)} dW(s). Since this is a deterministic function of Gaussian random variables (the Itô integral of a deterministic function is Gaussian), X(t) is Gaussian with mean E[X(t)] = X(0)e^{-θt} and variance Var(X(t)) = σ²∫₀ᵗ e^{-2θ(t-s)} ds = (σ²/2θ)(1 - e^{-2θt}).

As t → ∞, the mean decays to zero and the variance converges to σ²/(2θ). The process forgets its initial condition exponentially fast (at rate θ) and settles into a stationary Gaussian distribution N(0, σ²/(2θ)). The autocorrelation of the stationary process is R(τ) = (σ²/2θ)e^{-θ|τ|} — exponentially decaying with correlation time 1/θ. This is a fundamental model in physics (velocity of a Brownian particle under friction, by Uhlenbeck and Ornstein's original 1930 paper), finance (the Vasicek interest rate model), and biology (fluctuations around a homeostatic set point).

The OU process occupies a special place in the taxonomy of stochastic processes: it is the unique continuous-time process that is simultaneously Gaussian, Markov, and stationary. Brownian motion is Gaussian and Markov but not stationary (variance grows). A stationary Gaussian process with a non-exponential covariance function loses the Markov property. The exponential covariance is the only one compatible with all three properties, and this pins down the OU process uniquely (up to location and scale parameters). This characterization theorem explains why the OU process appears as the natural building block in so many contexts.