Ornstein-Uhlenbeck process

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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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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

In mathematics, the Ornstein-Uhlenbeck process, also known as the mean-reverting process, is a stochastic process 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. This equation is solved by using a variation of parameters argument. 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(rt) = r0e − θt + μ(1 − e − θt).

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).\,

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 Ornstein-Uhlenbeck process (an example of a Gaussian process that has a bounded variance) admits a stationary probability distribution, in contrast to the Wiener process.

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

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