Ornstein–Uhlenbeck process

Not to be confused with Ornstein–Uhlenbeck operator.

Five simulations with θ = 1, σ = 1 and μ = 0.

A 3D simulation with θ = 1, σ = 3, μ = (0, 0, 0) and the initial position (10, 10, 10).

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics, the physical sciences, and evolutionary biology. Its original application in physics was as a model for the velocity of a massive Brownian particle under the influence of friction. It is named after Leonard Ornstein and George Eugene Uhlenbeck.

The Ornstein–Uhlenbeck process is a Gaussian, time-homogeneous Markov process. For $\theta >0$, it admits the invariant Gaussian distribution

$X\sim {\mathcal {N}}\left(\mu ,{\frac {\sigma ^{2}}{2\theta }}\right).$

The process is stationary when its initial value is distributed according to this invariant distribution. If instead it begins from a fixed value or another nonstationary initial distribution, the process is generally not stationary; its distribution approaches the invariant Gaussian distribution as time increases. Its restoring drift toward $\mu$ gives the process its characteristic mean-reverting behavior.

The process can be considered to be a modification of the random walk in continuous time, or Wiener process, in which the properties of the process have been changed so that there is a tendency of the walk to move back towards a central location, with a greater attraction when the process is further away from the center. The Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of the discrete-time AR(1) process.