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 θ > 0 {\displaystyle \theta >0} , it admits the invariant Gaussian distribution
X ∼ N ( μ , σ 2 2 θ ) . {\displaystyle 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 μ {\displaystyle \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.
Definition
The Ornstein–Uhlenbeck process x t {\displaystyle x_{t}} is defined by the following stochastic differential equation:
d x t = − θ x t d t + σ d W t {\displaystyle dx_{t}=-\theta \,x_{t}\,dt+\sigma \,dW_{t}}
where θ > 0 {\displaystyle \theta >0} and σ > 0 {\displaystyle \sigma >0} are parameters and W t {\displaystyle W_{t}} denotes the Wiener process. An additional term is sometimes added:
d x t = − θ ( x t − μ ) d t + σ d W t {\displaystyle dx_{t}=-\theta (x_{t}-\mu )\,dt+\sigma \,dW_{t}}
where μ {\displaystyle \mu } is a constant called the (long-term) mean. The Ornstein–Uhlenbeck process is sometimes also written as a Langevin equation of the form
d x t d t = − θ x t + σ η ( t ) {\displaystyle {\frac {dx_{t}}{dt}}=-\theta \,x_{t}+\sigma \,\eta (t)}
where η ( t ) {\displaystyle \eta (t)} , also known as white noise, stands in for the supposed derivative d W t / d t {\displaystyle dW_{t}/dt} of the Wiener process. However, d W t / d t {\displaystyle dW_{t}/dt} does not exist because the Wiener process is nowhere differentiable, and so the Langevin equation only makes sense if interpreted in distributional sense. In physics and engineering disciplines, it is a common representation for the Ornstein–Uhlenbeck process and similar stochastic differential equations by tacitly assuming that the noise term is a derivative of a differentiable (e.g. Fourier) interpolation of the Wiener process.
Fokker–Planck equation representation The Ornstein–Uhlenbeck process can also be described in terms of a probability density function, P ( x , t ) {\displaystyle P(x,t)} , which specifies the probability of finding the process in the state x {\displaystyle x} at time t {\displaystyle t} . This function satisfies the Fokker–Planck equation
… excerpt ends here. Continue reading the full article.
