OrnsteinUhlenbeck

The Ornstein-Uhlenbeck process is a continuous-time stochastic process that exhibits mean reversion and Gaussian fluctuations. It was introduced by Leonard Ornstein and George Eugene Uhlenbeck in 1930 to model the velocity of a Brownian particle under friction and random forcing. The process is the solution to a linear stochastic differential equation and serves as a fundamental example in the theory of diffusion processes.

Mathematically, it is defined by the stochastic differential equation dX_t = θ(μ − X_t) dt + σ dW_t, where θ > 0 is the rate of mean reversion, μ is the long-term mean, σ ≥ 0 the volatility, and W_t a standard Brownian motion. The drift term pulls X_t toward μ at rate θ, while the diffusion term introduces Gaussian noise.

The Ornstein-Uhlenbeck process is Gaussian and Markovian; for any initial value X_0, its finite-time distributions are multivariate normal. It has a unique stationary distribution N(μ, σ^2/(2 θ)), toward which it converges as t → ∞. The autocovariance is Cov(X_t, X_{t+s}) = (σ^2/(2 θ)) e^{−θ s}, reflecting an exponential decay of memory.

For simulation over a step Δ, an exact discrete-time form is X_{t+Δ} = X_t e^{−θ Δ} + μ (1 − e^{−θ Δ}) + η, with η ~ N(0, (σ^2/(2 θ))(1 − e^{−2 θ Δ})). Euler-Maruyama is often used for approximate simulations.

Applications of the OU process span physics, finance (notably the Vasicek model for interest rates), and neuroscience as a model of mean-reverting quantities. It is Gaussian and unbounded, so it does not enforce positivity; in settings where positivity is required, nonlinear variants like the CIR process may be preferred. Variants include multidimensional OU processes and OU bridges.