Ornstein–Uhlenbeck process

In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. 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. In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. 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 process is a stationary Gauss–Markov process, which means that it is a Gaussian process, a Markov process, and is temporally homogeneous. The Ornstein–Uhlenbeck process is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables. Over time, the process tends to drift towards its long-term mean: such a process is called mean-reverting. 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. The Ornstein–Uhlenbeck process x t {displaystyle x_{t}} is defined by the following stochastic differential equation: where θ > 0 {displaystyle heta >0} and σ > 0 {displaystyle sigma >0} are parameters and W t {displaystyle W_{t}} denotes the Wiener process.