Ornstein-Uhlenbeck process

In mathematics, the Ornstein-Uhlenbeck process (named after Leonard Ornstein and George Eugene Uhlenbeck), also known as the mean-reverting process, is a stochastic process rt given by the following stochastic differential equation:


 * $$dr_t = -\theta (r_t-\mu)\,dt + \sigma\, dW_t,\,$$

where &theta;, &mu; and &sigma; are parameters and Wt denotes the Wiener process.

The Ornstein-Uhlenbeck process is an example of a Gaussian process that has a bounded variance and admits a stationary probability distribution, in contrast to the Wiener process. The stationary (long-term) variance is given by


 * $$VAR(r_t)={\sigma ^2 \over 2\theta}$$

The Ornstein-Uhlenbeck process is the continuous-time analogue of the discrete-time AR(1) process.

[[Image:OrnsteinUhlenbeck3.png|thumb|450px|three sample paths of different OU-processes with θ = 1, μ = 1.2, σ = 0.3:

navy : initial value a = 0 (a.s.)

olive : initial value a = 2 (a.s.)

red : initial value normally distributed so that the process has invariant measure]]

Solution
This equation is solved by variation of parameters. Apply Itō's lemma to the function $$f(r_t, t) = r_t e^{\theta t}$$ to get


 * $$df(r_t,t) = \theta r_t e^{\theta t}\, dt + e^{\theta t}\, dr_t\,$$


 * $$ = e^{\theta t}\theta \mu \, dt + \sigma e^{\theta t}\, dW_t. \, $$

Integrating from 0 to t we get


 * $$ r_t e^{\theta t} = r_0 + \int_0^t e^{\theta s}\theta \mu \, ds + \int_0^t \sigma e^{\theta s}\, dW_s \, $$

whereupon we see


 * $$ r_t = r_0 e^{-\theta t} + \mu(1-e^{-\theta t}) + \int_0^t \sigma e^{\theta (s-t)}\, dW_s. \, $$

Thus, the first moment is given by (assuming that $$r_0$$ is a constant),


 * $$E(r_t)=r_0 e^{-\theta t}+\mu(1-e^{-\theta t}) \!\ $$

Denote $$s \wedge t = \min(s,t)$$ we can use the Itō isometry to calculate the covariance function by


 * $$\operatorname{cov}(r_s,r_t)= E[(r_s - E[r_s])(r_t - E[r_t])]$$


 * $$= E[\int_0^s \sigma e^{\theta (u-s)}\, dW_u \int_0^t \sigma  e^{\theta (v-t)}\, dW_v ]$$


 * $$= \sigma^2 e^{-\theta (s+t)}E[\int_0^s e^{\theta u}\, dW_u \int_0^t  e^{\theta v}\, dW_v ]$$


 * $$= \frac{\sigma^2}{2\theta} \, e^{-\theta (s+t)}(e^{2\theta (s \wedge t)}-1).\,$$

Alternative representation I
It is also possible (and often convenient) to represent $$r_t$$ (unconditionally) as a scaled time-transformed Wiener process:


 * $$ r_t=\mu+{\sigma\over\sqrt{2\theta}}W(e^{2\theta t})e^{-\theta t} $$

or conditionally (given $$r_0$$) as


 * $$ r_t=r_0 e^{-\theta t} +\mu (1-e^{-\theta t})+

{\sigma\over\sqrt{2\theta}}W(e^{2\theta t}-1)e^{-\theta t}. $$

The time integral of this process can be used to generate noise with a 1/f power spectrum.

Alternative representation II
If B is a Brownian motion, then


 * $$U_t = \exp(\beta t) B\left(\frac{1-e^{-2\beta t}}{2\beta}\right)$$

defines an OU process and solves the equation


 * $$dU_t = \beta U_t \, dt + d W_t$$

where $$W$$ is a Brownian motion.

Generalisations
It is possible to extend the OU processes to processes where the background driving process is a Levy process. These processes are widely studied by Ole Barndorff-Nielsen and others.