OrnsteinUhlenbeckprosess

The Ornstein-Uhlenbeck process, sometimes written as Ornstein-Uhlenbeck process or OrnsteinUhlenbeckprosess in some languages, is a continuous-time Gaussian Markov process used to model mean-reverting phenomena. It is defined by the stochastic differential equation dX_t = κ(μ − X_t) dt + σ dW_t, where κ > 0 is the rate of mean reversion, μ is the long-term mean, σ > 0 is the volatility, and W_t is a standard Wiener process. A common alternative parameterization is dX_t = a(b − X_t) dt + σ dW_t with a, b > 0 and b real.

For a process with initial value X_0, the mean and variance at time t are E[X_t] = μ + (X_0 − μ) e^{−κ t} and Var[X_t] = (σ^2/(2κ))(1 − e^{−2κ t}). As t grows large, the distribution converges to a stationary Normal distribution N(μ, σ^2/(2κ)). The process is Gaussian, Markov, and continuously indexed; its autocovariance is Cov(X_t, X_{t+s}) = (σ^2/(2κ)) e^{−κ s} for s ≥ 0.

Exact discretization over a short interval Δ > 0 is given by X_{t+Δ} = X_t e^{−κ Δ} + μ(1 − e^{−κ Δ}) + σ sqrt((1 − e^{−2κ Δ})/(2κ)) Z, where Z is standard normal. This enables precise simulation without step-size bias.

Applications include physics, where it describes the velocity of a Brownian particle under friction, and finance, where the Vasicek model uses the Ornstein-Uhlenbeck form to model mean-reverting interest rates and other quantities. The process was developed in the early 20th century by Leonard Ornstein and George Eugene Uhlenbeck and remains a foundational model for mean-reverting stochastic dynamics.