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{{distinguish|Ornstein–Uhlenbeck operator}}
{{More footnotes|date=January 2011}}

In mathematics, the '''Ornstein–Uhlenbeck process''' (named after [[Leonard Ornstein]] and [[George Eugene Uhlenbeck]]), is a [[stochastic process]] that, roughly speaking, describes the velocity of a massive [[Brownian motion|Brownian particle]] under the influence of friction. The process is [[stationary process|stationary]], [[Gaussian process|Gaussian]], and [[Markov process|Markov]]ian, and is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables.<ref name=Doob>{{harvnb|Doob|1942}}</ref> Over time, the process tends to drift towards its long-term mean: such a process is called '''[[Mean reversion (finance)|mean-reverting]]'''.

The process can be considered to be a modification of the [[random walk]] in [[continuous time]], or [[Wiener process]], in which the properties of the process have been changed so that there is a tendency of the walk to move back towards a central location, with a greater attraction when the process is further away from the centre. The Ornstein–Uhlenbeck process can also be considered as the [[continuous time|continuous-time]] analogue of the [[discrete time|discrete-time]] [[Autoregressive|AR(1) process]].

== Representation via a stochastic differential equation ==

An Ornstein–Uhlenbeck process, ''x''''t'', satisfies the following [[stochastic differential equation]]:

:<math>dx_t = \theta (\mu-x_t)\,dt + \sigma\, dW_t</math>

where <math>\theta > 0</math>, <math>\mu</math> and <math>\sigma > 0</math> are parameters and <math>W_t</math> denotes the [[Wiener process]].

The above representation can be taken as the primary definition of an Ornstein–Uhlenbeck process.<ref name=Doob/>{{Citation needed|date=June 2011}}.

Making the long term mean stochastic to another SDE is a simplified version of the [[cointelation]] SDE.<ref name="wilmottM.com">{{cite journal|authors=Mahdavi Damghani B.|title=The Non-Misleading Value of Inferred Correlation: An Introduction to the Cointelation Model|journal=Wilmott Magazine|year=2013|url=http://onlinelibrary.wiley.com/doi/10.1002/wilm.10252/abstract}}</ref>

== Fokker–Planck equation representation ==

The probability density function ''ƒ''(''x'', ''t'') of the Ornstein–Uhlenbeck process satisfies the [[Fokker–Planck equation]]

: <math>\frac{\partial f}{\partial t} = \theta \frac{\partial}{\partial x} [(x - \mu) f] + \frac{\sigma^2}{2} \frac{\partial^2 f}{\partial x^2}</math>

The Green function of this linear parabolic partial differential equation, taking <math>\mu = 0</math> and <math>D = \sigma^2/2</math> for simplicity, and the initial condition consisting of a unit point mass at location <math>y</math> is

: <math>f(x,t) = \sqrt{\frac{\theta}{2 \pi D (1-e^{-2\theta t})}} \exp\left\{\frac{-\theta}{2D}\left[\frac{(x - y e^{-\theta t})^2}{1-e^{-2\theta t}}\right]\right\}</math>

The stationary solution of this equation is the limit for time tending to infinity which is a [[Gaussian distribution]] with mean <math>\mu</math> and variance
<math>\sigma^2/(2\theta)</math>

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

== Application in physical sciences ==
The Ornstein–Uhlenbeck process is a prototype of a noisy [[Relaxation (physics)|relaxation process]].
Consider for example a [[Hooke's law|Hookean spring]] with spring constant <math>k</math> whose dynamics is highly overdamped
with friction coefficient <math>\gamma</math>.
In the presence of thermal fluctuations with [[temperature]] <math>T</math>, the length <math>x(t)</math>
of the spring will fluctuate stochastically around the spring rest length <math>x_0</math>;
its stochastic dynamic is described by an Ornstein–Uhlenbeck process with:

: <math>
\begin{align}
\theta &=k/\gamma, \\
\mu & =x_0, \\
\sigma &=\sqrt{2k_B T/\gamma},
\end{align}
</math>

where <math>\sigma</math> is derived from the [[Stokes–Einstein equation]] <math>D=\sigma^2/2=k_B T/\gamma</math> for the effective diffusion constant.

In physical sciences, the stochastic differential equation of an Ornstein–Uhlenbeck process is rewritten as a [[Langevin equation]]
:<math> \gamma\dot{x}(t) = - k( x(t) - x_0 ) + \xi(t) </math>
where <math>\xi(t)</math> is [[Gaussian noise|white Gaussian noise]] with
<math>\langle\xi(t_1)\xi(t_2)\rangle = 2 k_B T\,\gamma\, \delta(t_1-t_2).</math>

At equilibrium, the spring stores an average energy <math> \langle E\rangle = k \langle (x-x_0)^2 \rangle /2=k_B T/2</math> in accordance with the [[equipartition theorem]].

== 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 <math>\mu</math> represents the equilibrium or mean value supported by fundamentals; <math>\sigma</math> the degree of volatility around it caused by shocks, and <math>\theta</math> the rate by which these shocks dissipate and the variable reverts towards the mean. One application of the process is a trading strategy known as [[pairs trade]].<ref>[http://www.cs.sunysb.edu/~skiena/691/lectures/lecture23.pdf Advantages of Pair Trading: Market Neutrality]</ref><ref>[http://www.ms.unimelb.edu.au/publications/RampertshammerStefan.pdf An Ornstein-Uhlenbeck Framework for Pairs Trading]</ref>

== Mathematical properties ==
The Ornstein–Uhlenbeck process is an example of a [[Gaussian process]] that has a bounded variance and admits a [[stationary process|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

:<math>\operatorname{var}(x_t)={\sigma ^2 \over 2\theta}. \, </math>

The Ornstein–Uhlenbeck process is the [[continuous time|continuous-time]] analogue of the [[discrete time|discrete-time]] [[Autoregressive|AR(1) process]].

[[Image:OrnsteinUhlenbeck4.png|thumb|450px|three sample paths of different OU-processes with ''θ'' = 1, ''μ'' = 1.2, ''σ'' = 0.3: 
'''blue''': initial value ''a'' = 0 ([[almost surely|a.s.]]) 
'''green''': initial value ''a'' = 2 (a.s.) 
'''red''': initial value normally distributed so that the process has invariant measure]]

== Solution ==

This stochastic differential equation is solved by [[variation of parameters]].{{Citation needed|date=June 2011}} Apply [[Itō's lemma]] to the function

: <math> f(x_t, t) = x_t e^{\theta t} \, </math>

to get

:<math>
\begin{align}
df(x_t,t) & = \theta x_t e^{\theta t}\, dt + e^{\theta t}\, dx_t \\[6pt]
& = e^{\theta t}\theta \mu \, dt + \sigma e^{\theta t}\, dW_t.
\end{align}
</math>

Integrating from 0 to ''t'' we get

:<math> x_t e^{\theta t} = x_0 + \int_0^t e^{\theta s}\theta \mu \, ds + \int_0^t \sigma e^{\theta s}\, dW_s \, </math>

whereupon we see

:<math> x_t = x_0 e^{-\theta t} + \mu(1-e^{-\theta t}) + \int_0^t \sigma e^{\theta (s-t)}\, dW_s. \, </math>

=== Formulas for moments of nonstationary processes ===
From this representation, the first [[moment (mathematics)|moment]] is given by (assuming that ''x''0 is a constant)

:<math>E(x_t)=x_0 e^{-\theta t}+\mu(1-e^{-\theta t}) \!\ </math>

The [[Itō isometry]] can be used to calculate the [[covariance function]] by

: <math>
\begin{align}
\operatorname{cov}(x_s,x_t) & = E[(x_s - E[x_s])(x_t - E[x_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}
</math>

Thus if ''s'' < ''t'' (so that min(''s'', ''t'') = ''s''), then we have

: <math> \operatorname{cov}(x_s,x_t) = \frac{\sigma^2}{2\theta}\left( e^{-\theta(t-s)} - e^{-\theta(t+s)} \right). </math>

== Alternative representation for nonstationary processes ==

It is also possible (and often convenient) to represent ''x''''t'' (unconditionally, i.e. as <math>t\rightarrow\infty</math>) as a scaled time-transformed Wiener process {{Citation needed|date=January 2012}}:

:<math> x_t=\mu+{\sigma\over\sqrt{2\theta}}e^{-\theta t}W_{e^{2\theta t}} </math>

or conditionally (given ''x''0) as

:<math> x_t=x_0 e^{-\theta t} +\mu (1-e^{-\theta t})+
{\sigma\over\sqrt{2\theta}}e^{-\theta t}W_{e^{2\theta t}-1}. </math>

The time integral of this process can be used to generate [[pink noise|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 <math>n</math> blue and yellow balls. At each step a ball is chosen at random and replaced by a ball of the opposite colour (equivalently, a ball chosen uniformly at random changes color). Let <math>X_n</math> be the number of blue balls in the urn after <math>n</math> steps. Then <math>\frac{X_{[nt]} - n/2}{\sqrt{n}}</math> converges in law to a Ornstein–Uhlenbeck process as <math>n</math> tends to infinity.

== Generalizations ==
It is possible to extend Ornstein–Uhlenbeck processes to processes where the background driving process is a [[Lévy process]].{{clarify|reason=how is this different|date=July 2011}} These processes are widely studied by [[Ole Barndorff-Nielsen]] and [[Neil Shephard]],{{citation needed|date= July 2011}} and others.{{citation needed|date=July 2011}}

In addition, in finance, stochastic processes are used the volatility increases for larger
values of <math>X</math>. In particular, the CKLS (Chan–Karolyi–Longstaff–Sanders) process<ref>Chan et al. (1992)</ref> with the volatility term replaced by <math>\sigma\,x^\gamma\, dW_t</math> can be solved in closed form for <math>\gamma=1/2</math> or 1, as well as for <math>\gamma=0</math>, which corresponds to the conventional OU process.

== See also ==
* [[Cointelation]]
* The [[Vasicek model]] of [[interest rates]] is an example of an Ornstein–Uhlenbeck process.
* [[Short rate model]] – contains more examples.

== Notes ==
{{Reflist}}

== References ==
*{{cite journal |first=E. |last=Bibbona |first2=G. |last2=Panfilo |first3=P. |last3=Tavella |title=The Ornstein-Uhlenbeck process as a model of a low pass filtered white noise |journal=Metrologia |volume=45 |issue=6 |pages=S117–S126 |year=2008 |doi=10.1088/0026-1394/45/6/S17 }}
*{{cite journal |last=Chan |first=K. C. |last2=Karolyi |first2=G. A. |last3=Longstaff |first3=F. A. |last4=Sanders |first4=A. B. |title=An empirical comparison of alternative models of the short-term interest rate |journal=[[Journal of Finance]] |volume=47 |issue=3 |pages=1209–1227 |year=1992 |doi=10.1111/j.1540-6261.1992.tb04011.x }}
*{{cite journal |first=J. L. |last=Doob |authorlink=Joseph Leo Doob |title=The Brownian movement and stochastic equations |journal=[[Annals of Mathematics]] |volume=43 |issue=2 |year=1942 |pages=351–369 |doi= |jstor=1968873 }}
*{{cite journal |first=D. T. |last=Gillespie |title=Exact numerical simulation of the Ornstein–Uhlenbeck process and its integral |journal=[[Physical Review|Phys. Rev. E]] |volume=54 |issue=2 |pages=2084–2091 |year=1996 |pmid=9965289 |doi=10.1103/PhysRevE.54.2084 }}
*{{cite book |first=H. |last=Risken |title=The Fokker–Planck Equation: Method of Solution and Applications |publisher=Springer-Verlag |location=New York |year=1989 |isbn=0387504982 }}
*{{cite journal |first=G. E. |last=Uhlenbeck |first2=L. S. |last2=Ornstein |title=On the theory of Brownian Motion |journal=Phys. Rev. |volume=36 |issue= |pages=823–841 |year=1930 |doi=10.1103/PhysRev.36.823 }}

== External links ==
*[http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1109160 A Stochastic Processes Toolkit for Risk Management], Damiano Brigo, Antonio Dalessandro, Matthias Neugebauer and Fares Triki
*[http://www.sitmo.com/article/calibrating-the-ornstein-uhlenbeck-model/ Simulating and Calibrating the Ornstein–Uhlenbeck process], M.A. van den Berg
*[http://www.investmentscience.com/Content/howtoArticles/MLE_for_OR_mean_reverting.pdf Maximum likelihood estimation of mean reverting processes], Jose Carlos Garcia Franco

{{Stochastic processes}}

{{DEFAULTSORT:Ornstein-Uhlenbeck process}}
[[Category:Stochastic differential equations]]
[[Category:Stochastic processes]]
[[Category:Variants of random walks]]

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