Ornstein-Uhlenbeck process

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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 θ, μ and σ 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.


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
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

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[edit] Solution

This equation is solved by variation of parameters. Apply Itō's lemma to the function f(rt,t) = rteθt 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 r0 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).\,

[edit] Alternative representation I

It is also possible (and often convenient) to represent rt (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 r0) 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.

[edit] 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.

[edit] References

  • G.E.Uhlenbeck and L.S.Ornstein: "On the theory of Brownian Motion", Phys.Rev. 36:823-41, 1930
  • D.T.Gillespie: "Exact numerical simulation of the Ornstein-Uhlenbeck process and its integral", Phys.Rev.E 54:2084-91, 1996

[edit] See also

The Vasicek model of interest rates is an example of an Ornstein-Uhlenbeck process.

[edit] 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.

[edit] External links

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