Ornstein–Uhlenbeck process

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 , it admits the invariant Gaussian distribution 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 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.