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
In the stationary regime, the autocovariance function is Cov(X_t, X_{t+s}) = (σ^2/(2θ)) e^{−θ|s|}, reflecting exponential decay of
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
Applications span physics (velocity of a Brownian particle with friction), finance (Vasicek model for interest rates),