OrnsteinUhlenbeckprocessen

The Ornstein-Uhlenbeck process, denoted here as OrnsteinUhlenbeckprocessen, is a continuous-time Gaussian Markov process that satisfies the stochastic differential equation dX_t = θ(μ − X_t) dt + σ dW_t, with parameters θ > 0, μ ∈ R and σ ≥ 0. It is the prototypical mean-reverting process and arises as a solution to the Langevin equation modeling a particle under friction in a noisy environment. The process is the continuous-time analogue of a simple autoregression and is widely used to describe quantities that tend to drift toward a long-term average.

Properties of the process include Gaussian finite-dimensional distributions and the Markov property. For a given initial value X_0, the distribution of (X_t) is fully determined by X_0 and the parameters. The mean and variance evolve as E[X_t] = X_0 e^{−θ t} + μ(1 − e^{−θ t}) and Var[X_t] = (σ^2/(2θ))(1 − e^{−2θ t}). As t → ∞, X_t converges in distribution to the stationary normal distribution N(μ, σ^2/(2θ)).

In the stationary regime, the autocovariance function is Cov(X_t, X_{t+s}) = (σ^2/(2θ)) e^{−θ|s|}, reflecting exponential decay of memory. The process is ergodic and mixes toward its stationary distribution, making it suitable for modeling long-run equilibria with random fluctuations.

Solution form and interpretation: X_t = X_0 e^{−θ t} + μ(1 − e^{−θ t}) + σ ∫_0^t e^{−θ (t−u)} dW_u. This explicit representation highlights its linear drift toward μ and additive noise. Extensions include multidimensional versions dX_t = A(m − X_t) dt + Σ dW_t.

Applications span physics (velocity of a Brownian particle with friction), finance (Vasicek model for interest rates), chemistry and neuroscience, where mean reversion and Gaussian noise are appropriate.