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Ornstein–Uhlenbeck process
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Not to be confused withOrnstein–Uhlenbeck operator.
In mathematics, the Ornstein–Uhlenbeck process (named afterLeonard Ornstein andGeorge Eugene Uhlenbeck), is astochastic process that, roughly speaking, describes the velocity of a massiveBrownian particle under the influence of friction. The process isstationary,Gaussian, andMarkov, and is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables.[1] Over time, the process tends to drift towards its long-term mean: such a process is called mean-reverting.
The process x t satisfies the followingstochastic differential equation:

where θ > 0, μ and σ > 0 are parameters and Wt denotes theWiener process.
Contents
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1 Application in physical sciences2 Application in financial mathematics3 Mathematical properties4 Solution5 Alternative representation6 Scaling limit interpretation7 Fokker–Planck equation representation8 Generalizations9 See also10 References11 External links
[edit] Application in physical sciences
The Ornstein–Uhlenbeck process is a prototype of a noisyrelaxation process. Consider for example aHookean spring with spring constant k whose dynamics is highly overdamped with friction coefficient γ. In the presence of thermal fluctuations withtemperature T, the length x(t) of the spring will fluctuate stochastically around the spring rest length x0; its stochastic dynamic is described by an Ornstein–Uhlenbeck process with:

where σ is derived from theStokes-Einstein equation D = σ2 / 2 = kBT / γ for the effective diffusion constant.
In physical sciences, the stochastic differential equation of an Ornstein–Uhlenbeck process is rewritten as aLangevin equation

where ξ(t) iswhite Gaussian noise with.
At equilibrium, the spring stores an average energyin accordance with theequipartition theorem.
[edit] Application in financial mathematics
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. One application of the process is a trading strategypairs trade.[2][3]
[edit] Mathematical properties
The Ornstein–Uhlenbeck process is an example of aGaussian process that has a bounded variance and admits astationaryprobability distribution, in contrast to theWiener 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

The Ornstein–Uhlenbeck process is thecontinuous-time analogue of thediscrete-timeAR(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
[edit] Solution
This equation is solved byvariation of parameters. ApplyItō–Doeblin's formula to the function

to get

Integrating from 0 to t we get

whereupon we see

Thus, the firstmoment is given by (assuming that x0 is a constant)

We can use theItō isometry to calculate thecovariance function by

Thus if s < t (so that min(s, t) = s), then we have

[edit] Alternative representation
It is also possible (and often convenient) to represent xt (unconditionally, i.e. as) as a scaled time-transformed Wiener process:

or conditionally (given x0) as

The time integral of this process can be used to generatenoise with a 1/ƒ power spectrum.
[edit] Scaling limit interpretation
The Ornstein–Uhlenbeck process can be interpreted as ascaling limit of a discrete process, in the same way thatBrownian motion is a scaling limit ofrandom 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 Xn be the number of blue balls in the urn after n steps. Thenconverges to a Ornstein–Uhlenbeck process as n tends to infinity.
[edit] Fokker–Planck equation representation
The probability density function ƒ(x, t) of the Ornstein–Uhlenbeck process satisfies theFokker–Planck equation

The stationary solution of this equation is aGaussian distribution with mean μ and variance σ2 / (2θ)

[edit] Generalizations
It is possible to extend the OU processes to processes where the background driving process is aLévy process. These processes are widely studied byOle Barndorff-Nielsen andNeil Shephard and others.
In addition, processes are used in finance where the volatility increases for larger values of X. In particular, the CKLS (Chan-Karolyi-Longstaff-Sanders) process[4] with the volatility term replaced bycan be solved in closed form for γ = 1 / 2 or 1, as well as for γ = 0, which corresponds to the conventional OU process.
[edit] See also
TheVasicek model ofinterest rates is an example of an Ornstein–Uhlenbeck process.Short rate model – contains more examples.

This article includes alist of references, but its sources remain unclear because it has insufficientinline citations.
Please help toimprove this article by introducing more precise citationswhere appropriate. (January 2011)
[edit] References
^Doob 1942^Advantages of Pair Trading: Market Neutrality^An Ornstein-Uhlenbeck Framework for Pairs Trading^ Chan et al. (1992)
G.E.Uhlenbeck and L.S.Ornstein: "On the theory of Brownian Motion", Phys.Rev. 36:823–41, 1930.doi:10.1103/PhysRev.36.823 D.T.Gillespie: "Exact numerical simulation of the Ornstein–Uhlenbeck process and its integral", Phys.Rev.E 54:2084–91, 1996.PMID 9965289doi:10.1103/PhysRevE.54.2084 H. Risken: "The Fokker–Planck Equation: Method of Solution and Applications", Springer-Verlag, New York, 1989 E. Bibbona, G. Panfilo and P. Tavella: "The Ornstein-Uhlenbeck process as a model of a low pass filtered white noise", Metrologia 45:S117-S126, 2008doi:10.1088/0026-1394/45/6/S17 Chan. K. C., Karolyi, G. A., Longstaff, F. A. & Sanders, A. B.: "An empirical comparison of alternative models of the short-term interest rate", Journal of Finance 52:1209–27, 1992.Doob, J.L. (1942), "The Brownian movement and stochastic equations", Ann. of Math. 43: 351–369 .
[edit] External links
A Stochastic Processes Toolkit for Risk Management, Damiano Brigo, Antonio Dalessandro, Matthias Neugebauer and Fares TrikiSimulating and Calibrating the Ornstein–Uhlenbeck process, M.A. van den BergCalibrating the Ornstein-Uhlenbeck model, M.A. van den BergMaximum likelihood estimation of mean reverting processes, Jose Carlos Garcia Franco
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