Ornstein–Uhlenbeck process: Definition from Answers.com

Not to be confused with Ornstein–Uhlenbeck operator.

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 r_t given by the following stochastic differential equation:

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

where θ > 0, μ and σ > 0 are parameters and W_t denotes the Wiener process.

The Ornstein–Uhlenbeck process is one of several approaches used to model (with modifications) interest rates, currency exchange rates, and commodity prices stochastically. The parameter $μ$ represents the equilibrium or mean value supported by fundamentals; $σ$ the degree of volatility around it caused by shocks, and $θ$ the rate by which these shocks dissipate and the variable reverts towards the mean.

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 difference between the two is in their "drift" term. For the Wiener process the drift term is constant, whereas for the Ornstein–Uhlenbeck process it is dependent on the current value of the process: if the current value of the process is less than the (long-term) mean, the drift will be positive; if the current value of the process is greater than the (long-term) mean, the drift will be negative. In other words, the mean acts as an equilibrium level for the process. This gives the process its informative name, "mean-reverting." The stationary (long-term) variance is given by

$\operatorname{var}(r_t)={\sigma ^2 \over 2\theta}. \,$

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:
blue: initial value a = 0 (a.s.)
green: initial value a = 2 (a.s.)
red: initial value normally distributed so that the process has invariant measure

1 Solution
2 Alternative representation
3 Scaling limit interpretation
4 Fokker–Planck equation representation
5 Generalisations
6 See also
7 References
8 External links

Solution

This equation is solved by variation of parameters. Apply Itō–Doeblin's formula to the function

$f(r_t, t) = r_t e^{\theta t} \,$

to get

$\begin{align} df(r_t,t) & = \theta r_t e^{\theta t}\, dt + e^{\theta t}\, dr_t \\ & = e^{\theta t}\theta \mu \, dt + \sigma e^{\theta t}\, dW_t. \end{align}$

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}) \!\$

We can use the Itō isometry to calculate the covariance function by

$\begin{align} \operatorname{cov}(r_s,r_t) & = E[(r_s - E[r_s])(r_t - E[r_t])] \\ & = E \left[ \int_0^s \sigma e^{\theta (u-s)}\, dW_u \int_0^t \sigma e^{\theta (v-t)}\, dW_v \right] \\ & = \sigma^2 e^{-\theta (s+t)}E \left[ \int_0^s e^{\theta u}\, dW_u \int_0^t e^{\theta v}\, dW_v \right] \\ & = \frac{\sigma^2}{2\theta} \, e^{-\theta (s+t)}(e^{2\theta \min(s,t)}-1). \end{align}$

Alternative representation

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 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/ƒ power spectrum.

Scaling limit interpretation

The Ornstein–Uhlenbeck process can be interpreted as a scaling limit of a discrete process, in the same way that Brownian motion is a scaling limit of random walks. Consider an urn containing $n$ blue and yellow balls. At each step a ball is chosen at random and replaced by a ball of the opposite colour. Let $X n$ be the number of blue balls in the urn after $n$ steps. Then $\frac{X_{[nt]} - n/2}{\sqrt{n}}$ converges as $n$ tends to infinity to a Ornstein–Uhlenbeck process.

Fokker–Planck equation representation

The Fokker–Planck equation describing the distribution ƒ(x, t) of the OU process is given by

$\frac{\partial f}{\partial t} = \theta \frac{\partial}{\partial x} ((x - \mu) f) + \frac{\sigma^2}{2} \frac{\partial^2 f}{\partial x^2}.$

This equation has the stationary solution

$f_s(x) = \sqrt{\frac{\theta}{\pi \sigma^2}}\, e^{-\theta (x-\mu)^2/\sigma^2}.$

Generalisations

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

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
H. Risken: "The Fokker–Planck Equation: Method of Solution and Applications", Springer-Verlag, New York, 1989

External links

Simulating and Calibrating the Ornstein–Uhlenbeck process, sitmo.com

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Ornstein–Uhlenbeck process

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