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Ornstein–Uhlenbeck process

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Ornstein%E2%80%93Uhlenbeck_operator" title="Ornstein–Uhlenbeck operator">Ornstein–Uhlenbeck operator</a>.</div>

In mathematics, the Ornstein–Uhlenbeck process is a <a href="Stochastic_process" title="Stochastic process">stochastic process</a> with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive <a href="Brownian_motion" title="Brownian motion">Brownian particle</a> under the influence of <a href="Friction" title="Friction">friction</a>. It is named after <a href="Leonard_Ornstein" title="Leonard Ornstein">Leonard Ornstein</a> and <a href="George_Eugene_Uhlenbeck" class="mw-redirect" title="George Eugene Uhlenbeck">George Eugene Uhlenbeck</a>.

The Ornstein–Uhlenbeck process is a <a href="Stationary_process" title="Stationary process">stationary</a> <a href="Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov process</a>, which means that it is a <a href="Gaussian_process" title="Gaussian process">Gaussian process</a>, a <a href="Markov_process" class="mw-redirect" title="Markov process">Markov process</a>, and is temporally homogeneous. In fact, it is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables.<a href="#cite_note-FOOTNOTEDoob1942-1">[1]</a> Over time, the process tends to drift towards its mean function: such a process is called mean-reverting.

The process can be considered to be a modification of the <a href="Random_walk" title="Random walk">random walk</a> in <a href="Continuous_time" class="mw-redirect" title="Continuous time">continuous time</a>, or <a href="Wiener_process" title="Wiener process">Wiener process</a>, 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 center. The Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of the discrete-time <a href="Autoregressive" class="mw-redirect" title="Autoregressive">AR(1) process</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The Ornstein–Uhlenbeck process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{t}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{t}}</annotation>

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</math><img src="./f279a30bc8eabc788f3fe81c9cfb674e72e858db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.156ex; height:2.009ex;" alt="{\displaystyle x_{t}}" loading="lazy"> is defined by the following <a href="Stochastic_differential_equation" title="Stochastic differential equation">stochastic differential equation</a>:

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<annotation encoding="application/x-tex">{\displaystyle dx_{t}=-\theta \,x_{t}\,dt+\sigma \,dW_{t}}</annotation>

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</math><img src="./57faf896fb650bd968dc75a10af4d26ed53fa956.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.147ex; height:2.509ex;" alt="{\displaystyle dx_{t}=-\theta \,x_{t}\,dt+\sigma \,dW_{t}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta >0}">

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<annotation encoding="application/x-tex">{\displaystyle \theta >0}</annotation>

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</math><img src="./0b0ac07626379d065418cc158ce6be9aeccf33b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle \theta >0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma >0}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma >0}</annotation>

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</math><img src="./762ecd0f0905dd0d4d7a07f80fa8bfb324b9b021.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle \sigma >0}" loading="lazy"> are parameters and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{t}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{t}}</annotation>

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</math><img src="./50680c5535c83badfa630dba63b583d2eeaa2977.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.02ex; height:2.509ex;" alt="{\displaystyle W_{t}}" loading="lazy"> denotes the <a href="Wiener_process" title="Wiener process">Wiener process</a>.<a href="#cite_note-FOOTNOTEKaratzasShreve1991358-2">[2]</a><a href="#cite_note-FOOTNOTEGard1988115-3">[3]</a><a href="#cite_note-FOOTNOTEGardiner1985-4">[4]</a>

An additional term is sometimes added:

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<annotation encoding="application/x-tex">{\displaystyle dx_{t}=\theta (\mu -x_{t})\,dt+\sigma \,dW_{t}}</annotation>

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</math><img src="./906f3dc510ac425dbda85cc2e53bea0798be1abe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.003ex; height:2.843ex;" alt="{\displaystyle dx_{t}=\theta (\mu -x_{t})\,dt+\sigma \,dW_{t}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> is a constant called the (long-term) mean.

The Ornstein–Uhlenbeck process is sometimes also written as a <a href="Langevin_equation" title="Langevin equation">Langevin equation</a> of the form

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<annotation encoding="application/x-tex">{\displaystyle {\frac {dx_{t}}{dt}}=-\theta \,x_{t}+\sigma \,\eta (t)}</annotation>

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</math><img src="./24eb7c6cd63cd1b87c7b5249317847f82018a90f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.123ex; height:5.509ex;" alt="{\displaystyle {\frac {dx_{t}}{dt}}=-\theta \,x_{t}+\sigma \,\eta (t)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta (t)}">

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<annotation encoding="application/x-tex">{\displaystyle \eta (t)}</annotation>

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</math><img src="./f7976cedf52ec06ee04d45996ecfea61dbefee3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.818ex; height:2.843ex;" alt="{\displaystyle \eta (t)}" loading="lazy">, also known as <a href="White_noise" title="White noise">white noise</a>, stands in for the supposed derivative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dW_{t}/dt}">

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<annotation encoding="application/x-tex">{\displaystyle dW_{t}/dt}</annotation>

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</math><img src="./43a1bec6e3bcbbfebc1f6b1a941a5fca1e43d35e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.453ex; height:2.843ex;" alt="{\displaystyle dW_{t}/dt}" loading="lazy"> of the Wiener process.<a href="#cite_note-FOOTNOTERisken1989-5">[5]</a> However, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dW_{t}/dt}">

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<annotation encoding="application/x-tex">{\displaystyle dW_{t}/dt}</annotation>

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</math><img src="./43a1bec6e3bcbbfebc1f6b1a941a5fca1e43d35e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.453ex; height:2.843ex;" alt="{\displaystyle dW_{t}/dt}" loading="lazy"> does not exist because the Wiener process is nowhere differentiable,<a href="#cite_note-FOOTNOTELawler2006-6">[6]</a> and so the Langevin equation only makes sense if interpreted in distributional sense. In physics and engineering disciplines, it is a common representation for the Ornstein–Uhlenbeck process and similar stochastic differential equations by tacitly assuming that the noise term is a derivative of a differentiable (e.g. Fourier) interpolation of the Wiener process.

<div class="mw-heading mw-heading2"><h2 id="Fokker–Planck_equation_representation">Fokker–Planck equation representation</h2></div>

The Ornstein–Uhlenbeck process can also be described in terms of a probability density function, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x,t)}">

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<annotation encoding="application/x-tex">{\displaystyle P(x,t)}</annotation>

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</math><img src="./8567fcffb86758f24b5fabf8325b0c69879a8f08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.758ex; height:2.843ex;" alt="{\displaystyle P(x,t)}" loading="lazy">, which specifies the probability of finding the process in the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">.<a href="#cite_note-FOOTNOTERisken1989-5">[5]</a> This function satisfies the <a href="Fokker%E2%80%93Planck_equation" title="Fokker–Planck equation">Fokker–Planck equation</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial P}{\partial t}}=\theta {\frac {\partial }{\partial x}}(xP)+D{\frac {\partial ^{2}P}{\partial x^{2}}}}</annotation>

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</math><img src="./fe0333cf8b5b215fdf3ea9b1af7e11285aaf1471.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.201ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial P}{\partial t}}=\theta {\frac {\partial }{\partial x}}(xP)+D{\frac {\partial ^{2}P}{\partial x^{2}}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=\sigma ^{2}/2}">

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<annotation encoding="application/x-tex">{\displaystyle D=\sigma ^{2}/2}</annotation>

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</math><img src="./657c30593c183d872f0b006925f3266080c180fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.732ex; height:3.176ex;" alt="{\displaystyle D=\sigma ^{2}/2}" loading="lazy">. This is a linear <a href="Parabolic_partial_differential_equation" title="Parabolic partial differential equation">parabolic partial differential equation</a> which can be solved by a variety of techniques. The transition probability, also known as the <a href="Green's_function" title="Green's function">Green's function</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x,t\mid x',t')}">

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<annotation encoding="application/x-tex">{\displaystyle P(x,t\mid x',t')}</annotation>

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</math><img src="./5b2176f71d37b885747f530cc402146ee71ce193.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.268ex; height:3.009ex;" alt="{\displaystyle P(x,t\mid x',t')}" loading="lazy"> is a Gaussian with mean <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'e^{-\theta (t-t')}+\mu (1-e^{-\theta (t-t')})}">

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<annotation encoding="application/x-tex">{\displaystyle x'e^{-\theta (t-t')}+\mu (1-e^{-\theta (t-t')})}</annotation>

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</math><img src="./aefa3dfe39ea295d3c9e176c57b8e75df68f4e87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.353ex; height:3.343ex;" alt="{\displaystyle x'e^{-\theta (t-t')}+\mu (1-e^{-\theta (t-t')})}" loading="lazy"> and variance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {D}{\theta }}\left(1-e^{-2\theta (t-t')}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {D}{\theta }}\left(1-e^{-2\theta (t-t')}\right)}</annotation>

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</math><img src="./e9cab0bb3ea70991d0b9bc0e508bcef6603bf380.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.39ex; height:5.343ex;" alt="{\displaystyle {\frac {D}{\theta }}\left(1-e^{-2\theta (t-t')}\right)}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle P(x,t\mid x',t')={\sqrt {\frac {\theta }{2\pi D(1-e^{-2\theta (t-t')})}}}\exp \left[-{\frac {\theta }{2D}}{\frac {(x-x'e^{-\theta (t-t')}-\mu (1-e^{-\theta (t-t')}))^{2}}{1-e^{-2\theta (t-t')}}}\right]}</annotation>

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</math><img src="./1b4007fd321d948f775e25883eabca18ec78d5f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:85.825ex; height:7.843ex;" alt="{\displaystyle P(x,t\mid x',t')={\sqrt {\frac {\theta }{2\pi D(1-e^{-2\theta (t-t')})}}}\exp \left[-{\frac {\theta }{2D}}{\frac {(x-x'e^{-\theta (t-t')}-\mu (1-e^{-\theta (t-t')}))^{2}}{1-e^{-2\theta (t-t')}}}\right]}" loading="lazy"></dd></dl>

This gives the probability of the state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> occurring at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> given initial state <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}">

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</math><img src="./0ac74959896052e160a5953102e4bc3850fe93b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}" loading="lazy"> at time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t'<t}">

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</math><img src="./f82f38212f60cb24782a32da93e8279acc2ceb9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.462ex; height:2.509ex;" alt="{\displaystyle t'<t}" loading="lazy">. Equivalently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x,t\mid x',t')}">

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<annotation encoding="application/x-tex">{\displaystyle P(x,t\mid x',t')}</annotation>

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</math><img src="./5b2176f71d37b885747f530cc402146ee71ce193.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.268ex; height:3.009ex;" alt="{\displaystyle P(x,t\mid x',t')}" loading="lazy"> is the solution of the Fokker–Planck equation with initial condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x,t')=\delta (x-x')}">

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<annotation encoding="application/x-tex">{\displaystyle P(x,t')=\delta (x-x')}</annotation>

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</math><img src="./1a96bbfce396d3cdee407dd3869a81711ac4b88e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.584ex; height:3.009ex;" alt="{\displaystyle P(x,t')=\delta (x-x')}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Mathematical_properties">Mathematical properties</h2></div>

Conditioned on a particular value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

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</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy">, the mean is

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {\mathbb {E} } (x_{t}\mid x_{0})=x_{0}e^{-\theta t}+\mu (1-e^{-\theta t})}</annotation>

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</math><img src="./e2cfb1e2556203b6df08b5b99d27d8dca30200f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.291ex; height:3.176ex;" alt="{\displaystyle \operatorname {\mathbb {E} } (x_{t}\mid x_{0})=x_{0}e^{-\theta t}+\mu (1-e^{-\theta t})}" loading="lazy"></dd></dl>

and the <a href="Covariance" title="Covariance">covariance</a> is

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {cov} (x_{s},x_{t})={\frac {\sigma ^{2}}{2\theta }}\left(e^{-\theta |t-s|}-e^{-\theta (t+s)}\right).}</annotation>

</semantics>

</math><img src="./315e9e6f29658f6519e27cc5c8c6f232200717b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:38.322ex; height:5.843ex;" alt="{\displaystyle \operatorname {cov} (x_{s},x_{t})={\frac {\sigma ^{2}}{2\theta }}\left(e^{-\theta |t-s|}-e^{-\theta (t+s)}\right).}" loading="lazy"></dd></dl>

For the stationary (unconditioned) process, the mean of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{t}}">

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</math><img src="./f279a30bc8eabc788f3fe81c9cfb674e72e858db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.156ex; height:2.009ex;" alt="{\displaystyle x_{t}}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy">, and the covariance of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{s}}">

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</math><img src="./731c17e8ce0dad32aab6e3ad68f52405fe277007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.333ex; height:2.009ex;" alt="{\displaystyle x_{s}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{t}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{t}}</annotation>

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</math><img src="./f279a30bc8eabc788f3fe81c9cfb674e72e858db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.156ex; height:2.009ex;" alt="{\displaystyle x_{t}}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\sigma ^{2}}{2\theta }}e^{-\theta |t-s|}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\sigma ^{2}}{2\theta }}e^{-\theta |t-s|}}</annotation>

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</math><img src="./b28ff5d160cffdbe16c2f5f5b396b83e7cd1a186.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.144ex; height:5.843ex;" alt="{\displaystyle {\frac {\sigma ^{2}}{2\theta }}e^{-\theta |t-s|}}" loading="lazy">.

The Ornstein–Uhlenbeck process is an example of a <a href="Gaussian_process" title="Gaussian process">Gaussian process</a> that has a bounded variance and admits a <a href="Stationary_process" title="Stationary process">stationary</a> <a href="Probability_distribution" title="Probability distribution">probability distribution</a>, in contrast to the <a href="Wiener_process" title="Wiener process">Wiener process</a>; 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."

<div class="mw-heading mw-heading3"><h3 id="Properties_of_sample_paths">Properties of sample paths</h3></div>

A temporally homogeneous Ornstein–Uhlenbeck process can be represented as a scaled, time-transformed <a href="Wiener_process" title="Wiener process">Wiener process</a>:

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<annotation encoding="application/x-tex">{\displaystyle x_{t}={\frac {\sigma }{\sqrt {2\theta }}}e^{-\theta t}W_{e^{2\theta t}}}</annotation>

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</math><img src="./4d0b1d02e953950d3938618bed21f1e504f488fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.37ex; height:5.676ex;" alt="{\displaystyle x_{t}={\frac {\sigma }{\sqrt {2\theta }}}e^{-\theta t}W_{e^{2\theta t}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{t}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{t}}</annotation>

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</math><img src="./50680c5535c83badfa630dba63b583d2eeaa2977.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.02ex; height:2.509ex;" alt="{\displaystyle W_{t}}" loading="lazy"> is the standard Wiener process. This is roughly Theorem 1.2 in <a href="#CITEREFDoob1942">Doob 1942</a>. Equivalently, with the change of variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=e^{2\theta t}}">

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<annotation encoding="application/x-tex">{\displaystyle s=e^{2\theta t}}</annotation>

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</math><img src="./47618c0169be55ff3315b5a91757cae6957b7e29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.691ex; height:2.676ex;" alt="{\displaystyle s=e^{2\theta t}}" loading="lazy"> this becomes

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<annotation encoding="application/x-tex">{\displaystyle W_{s}={\frac {\sqrt {2\theta }}{\sigma }}s^{1/2}x_{(\ln s)/(2\theta )},\qquad s>0}</annotation>

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</math><img src="./92f5b87f1c0a9724bcc4add28bb54743c964696e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.205ex; height:5.843ex;" alt="{\displaystyle W_{s}={\frac {\sqrt {2\theta }}{\sigma }}s^{1/2}x_{(\ln s)/(2\theta )},\qquad s>0}" loading="lazy"></dd></dl>

Using this mapping, one can translate known properties of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{t}}">

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</math><img src="./50680c5535c83badfa630dba63b583d2eeaa2977.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.02ex; height:2.509ex;" alt="{\displaystyle W_{t}}" loading="lazy"> into corresponding statements for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{t}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{t}}</annotation>

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</math><img src="./f279a30bc8eabc788f3fe81c9cfb674e72e858db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.156ex; height:2.009ex;" alt="{\displaystyle x_{t}}" loading="lazy">. For instance, the <a href="Wiener_process#Law_of_the_iterated_logarithm" title="Wiener process">law of the iterated logarithm</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{t}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle W_{t}}</annotation>

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</math><img src="./50680c5535c83badfa630dba63b583d2eeaa2977.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.02ex; height:2.509ex;" alt="{\displaystyle W_{t}}" loading="lazy"> becomes<a href="#cite_note-FOOTNOTEDoob1942-1">[1]</a>

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<mtext>with probability 1.</mtext>

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<annotation encoding="application/x-tex">{\displaystyle \limsup _{t\to \infty }{\frac {x_{t}}{\sqrt {(\sigma ^{2}/\theta )\ln t}}}=1,\quad {\text{with probability 1.}}}</annotation>

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</math><img src="./abb48e9a92ddab28c2414b66fea19852477926a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.963ex; height:6.009ex;" alt="{\displaystyle \limsup _{t\to \infty }{\frac {x_{t}}{\sqrt {(\sigma ^{2}/\theta )\ln t}}}=1,\quad {\text{with probability 1.}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Formal_solution">Formal solution</h3></div>

The stochastic differential equation for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{t}}">

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</math><img src="./f279a30bc8eabc788f3fe81c9cfb674e72e858db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.156ex; height:2.009ex;" alt="{\displaystyle x_{t}}" loading="lazy"> can be formally solved by <a href="Variation_of_parameters" title="Variation of parameters">variation of parameters</a>.<a href="#cite_note-FOOTNOTEGardiner1985106-7">[7]</a> Writing

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<annotation encoding="application/x-tex">{\displaystyle f(x_{t},t)=x_{t}e^{\theta t}\,}</annotation>

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</math><img src="./aae6bf82809046c1bcace63cbb89a9e60a32c7a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.439ex; height:3.176ex;" alt="{\displaystyle f(x_{t},t)=x_{t}e^{\theta t}\,}" loading="lazy"></dd></dl>

we get

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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{aligned}}}</annotation>

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</math><img src="./871e0ae642c2661631c788771dbe36e9b0a29027.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.692ex; height:7.843ex;" alt="{\displaystyle {\begin{aligned}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{aligned}}}" loading="lazy"></dd></dl>

Integrating from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> we get

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<annotation encoding="application/x-tex">{\displaystyle 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}\,}</annotation>

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</math><img src="./64d81e1b12a6e909eec134b42cb92fb228fbc06b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.546ex; height:6.176ex;" alt="{\displaystyle 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}\,}" loading="lazy"></dd></dl>

whereupon we see

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<annotation encoding="application/x-tex">{\displaystyle x_{t}=x_{0}\,e^{-\theta t}+\mu \,(1-e^{-\theta t})+\sigma \int _{0}^{t}e^{-\theta (t-s)}\,dW_{s}.\,}</annotation>

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</math><img src="./ac5c13f4e6769fce67cca4b29194198e90869be4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:47.741ex; height:6.176ex;" alt="{\displaystyle x_{t}=x_{0}\,e^{-\theta t}+\mu \,(1-e^{-\theta t})+\sigma \int _{0}^{t}e^{-\theta (t-s)}\,dW_{s}.\,}" loading="lazy"></dd></dl>

From this representation, the first <a href="Moment_(mathematics)" title="Moment (mathematics)">moment</a> (i.e. the mean) is shown to be

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (x_{t})=x_{0}e^{-\theta t}+\mu (1-e^{-\theta t})\!\ }</annotation>

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</math><img src="./5528045b5055b05e719ee468388715b1cff4d102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.196ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} (x_{t})=x_{0}e^{-\theta t}+\mu (1-e^{-\theta t})\!\ }" loading="lazy"></dd></dl>

assuming <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

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</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> is constant. Moreover, the <a href="It%C5%8D_isometry" class="mw-redirect" title="Itō isometry">Itō isometry</a> can be used to calculate the <a href="Covariance_function" title="Covariance function">covariance function</a> by

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {cov} (x_{s},x_{t})&=\operatorname {E} [(x_{s}-\operatorname {E} [x_{s}])(x_{t}-\operatorname {E} [x_{t}])]\\[5pt]&=\operatorname {E} \left[\int _{0}^{s}\sigma e^{\theta (u-s)}\,dW_{u}\int _{0}^{t}\sigma e^{\theta (v-t)}\,dW_{v}\right]\\[5pt]&=\sigma ^{2}e^{-\theta (s+t)}\operatorname {E} \left[\int _{0}^{s}e^{\theta u}\,dW_{u}\int _{0}^{t}e^{\theta v}\,dW_{v}\right]\\[5pt]&={\frac {\sigma ^{2}}{2\theta }}\,e^{-\theta (s+t)}(e^{2\theta \min(s,t)}-1)\\[5pt]&={\frac {\sigma ^{2}}{2\theta }}\left(e^{-\theta |t-s|}-e^{-\theta (t+s)}\right).\end{aligned}}}</annotation>

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</math><img src="./4586b8798608d0e84e69a03c3e8fe278fb48d4ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.671ex; width:52.327ex; height:32.509ex;" alt="{\displaystyle {\begin{aligned}\operatorname {cov} (x_{s},x_{t})&=\operatorname {E} [(x_{s}-\operatorname {E} [x_{s}])(x_{t}-\operatorname {E} [x_{t}])]\\[5pt]&=\operatorname {E} \left[\int _{0}^{s}\sigma e^{\theta (u-s)}\,dW_{u}\int _{0}^{t}\sigma e^{\theta (v-t)}\,dW_{v}\right]\\[5pt]&=\sigma ^{2}e^{-\theta (s+t)}\operatorname {E} \left[\int _{0}^{s}e^{\theta u}\,dW_{u}\int _{0}^{t}e^{\theta v}\,dW_{v}\right]\\[5pt]&={\frac {\sigma ^{2}}{2\theta }}\,e^{-\theta (s+t)}(e^{2\theta \min(s,t)}-1)\\[5pt]&={\frac {\sigma ^{2}}{2\theta }}\left(e^{-\theta |t-s|}-e^{-\theta (t+s)}\right).\end{aligned}}}" loading="lazy"></dd></dl>

Since the Itô integral of deterministic integrand is normally distributed, it follows that

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<annotation encoding="application/x-tex">{\displaystyle x_{t}=x_{0}e^{-\theta t}+\mu (1-e^{-\theta t})+{\tfrac {\sigma }{\sqrt {2\theta }}}W_{1-e^{-2\theta t}}}</annotation>

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</math><img src="./c9f84802c18362e8ad7f7ae10e312a4f443029be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.519ex; height:4.176ex;" alt="{\displaystyle x_{t}=x_{0}e^{-\theta t}+\mu (1-e^{-\theta t})+{\tfrac {\sigma }{\sqrt {2\theta }}}W_{1-e^{-2\theta t}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Kolmogorov_equations">Kolmogorov equations</h3></div>

The <a href="Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">infinitesimal generator</a> of the process is<a href="#cite_note-8">[8]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Lf=-\theta (x-\mu )f'+{\frac {1}{2}}\sigma ^{2}f''}">

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<annotation encoding="application/x-tex">{\displaystyle Lf=-\theta (x-\mu )f'+{\frac {1}{2}}\sigma ^{2}f''}</annotation>

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</math>If we let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=(x-\mu ){\sqrt {\frac {2\theta }{\sigma ^{2}}}}}">

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<annotation encoding="application/x-tex">{\displaystyle y=(x-\mu ){\sqrt {\frac {2\theta }{\sigma ^{2}}}}}</annotation>

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</math><img src="./e2cb7a31bcdf1b9d80b0d9e6c13960e8dfb22320.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.18ex; height:6.343ex;" alt="{\displaystyle y=(x-\mu ){\sqrt {\frac {2\theta }{\sigma ^{2}}}}}" loading="lazy">, then the eigenvalue equation simplifies to:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}}{dy^{2}}}\phi -y{\frac {d}{dy}}\phi -{\frac {\lambda }{\theta }}\phi =0}</annotation>

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</math>which is the defining equation for <a href="Hermite_polynomials" title="Hermite polynomials">Hermite polynomials</a>. Its solutions are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (y)=He_{n}(y)}">

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<annotation encoding="application/x-tex">{\displaystyle \phi (y)=He_{n}(y)}</annotation>

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</math><img src="./aa1dda9e1d554bc6cf3a9e0012c0eee61202f79f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.779ex; height:2.843ex;" alt="{\displaystyle \phi (y)=He_{n}(y)}" loading="lazy">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =-n\theta }">

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<annotation encoding="application/x-tex">{\displaystyle \lambda =-n\theta }</annotation>

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</math><img src="./b7dadceddde65577b961a59f55f50150e7a78ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.747ex; height:2.343ex;" alt="{\displaystyle \lambda =-n\theta }" loading="lazy">, which implies that the mean first passage time for a particle to hit a point on the boundary is on the order of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle \theta ^{-1}}</annotation>

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</math><img src="./973ca2ab2ebddcbb2ffdb73ef3e9b1cb49b29324.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.423ex; height:2.676ex;" alt="{\displaystyle \theta ^{-1}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Numerical_simulation">Numerical simulation</h2></div>

By using discretely sampled data at time intervals of width <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

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</math><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy">, the <a href="Maximum_likelihood_estimation" title="Maximum likelihood estimation">maximum likelihood estimators</a> for the parameters of the Ornstein–Uhlenbeck process are asymptotically normal to their true values.<a href="#cite_note-FOOTNOTEAït-Sahalia2002223–262-9">[9]</a> More precisely,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}\left({\begin{pmatrix}{\widehat {\theta }}_{n}\\{\widehat {\mu }}_{n}\\{\widehat {\sigma }}_{n}^{2}\end{pmatrix}}-{\begin{pmatrix}\theta \\\mu \\\sigma ^{2}\end{pmatrix}}\right)\xrightarrow {d} \ {\mathcal {N}}\left({\begin{pmatrix}0\\0\\0\end{pmatrix}},{\begin{pmatrix}{\frac {e^{2t\theta }-1}{t^{2}}}&0&{\frac {\sigma ^{2}(e^{2t\theta }-1-2t\theta )}{t^{2}\theta }}\\0&{\frac {\sigma ^{2}\left(e^{t\theta }+1\right)}{2\left(e^{t\theta }-1\right)\theta }}&0\\{\frac {\sigma ^{2}(e^{2t\theta }-1-2t\theta )}{t^{2}\theta }}&0&{\frac {\sigma ^{4}\left[\left(e^{2t\theta }-1\right)^{2}+2t^{2}\theta ^{2}\left(e^{2t\theta }+1\right)+4t\theta \left(e^{2t\theta }-1\right)\right]}{t^{2}\left(e^{2t\theta }-1\right)\theta ^{2}}}\end{pmatrix}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}\left({\begin{pmatrix}{\widehat {\theta }}_{n}\\{\widehat {\mu }}_{n}\\{\widehat {\sigma }}_{n}^{2}\end{pmatrix}}-{\begin{pmatrix}\theta \\\mu \\\sigma ^{2}\end{pmatrix}}\right)\xrightarrow {d} \ {\mathcal {N}}\left({\begin{pmatrix}0\\0\\0\end{pmatrix}},{\begin{pmatrix}{\frac {e^{2t\theta }-1}{t^{2}}}&0&{\frac {\sigma ^{2}(e^{2t\theta }-1-2t\theta )}{t^{2}\theta }}\\0&{\frac {\sigma ^{2}\left(e^{t\theta }+1\right)}{2\left(e^{t\theta }-1\right)\theta }}&0\\{\frac {\sigma ^{2}(e^{2t\theta }-1-2t\theta )}{t^{2}\theta }}&0&{\frac {\sigma ^{4}\left[\left(e^{2t\theta }-1\right)^{2}+2t^{2}\theta ^{2}\left(e^{2t\theta }+1\right)+4t\theta \left(e^{2t\theta }-1\right)\right]}{t^{2}\left(e^{2t\theta }-1\right)\theta ^{2}}}\end{pmatrix}}\right)}</annotation>

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</math>

To simulate an OU process numerically with standard deviation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

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</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> and correlation time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =1/\Theta }">

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<annotation encoding="application/x-tex">{\displaystyle \tau =1/\Theta }</annotation>

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</math><img src="./355a271b148dd7f87111240d6d228d001c576a68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.433ex; height:2.843ex;" alt="{\displaystyle \tau =1/\Theta }" loading="lazy">, one method is to apply the finite-difference formula

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<annotation encoding="application/x-tex">{\displaystyle x(t+dt)=x(t)-\Theta \,dt\,x(t)+\Sigma {\sqrt {2\,dt\,\Theta }}\nu _{i}}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu _{i}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \nu _{i}}</annotation>

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</math><img src="./dcb0577728049599bceabd5ed148f426e9d44308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.948ex; height:2.009ex;" alt="{\displaystyle \nu _{i}}" loading="lazy"> is a normally distributed random number with zero mean and unit variance, sampled independently at every time-step <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dt}">

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<annotation encoding="application/x-tex">{\displaystyle dt}</annotation>

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</math><img src="./ebee76a835701fd1f26047a09855f2ea36bb08fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.055ex; height:2.176ex;" alt="{\displaystyle dt}" loading="lazy">.<a href="#cite_note-FOOTNOTEKloedenPlatenSchurz1994-10">[10]</a>

<div class="mw-heading mw-heading2"><h2 id="Scaling_limit_interpretation">Scaling limit interpretation</h2></div>

The Ornstein–Uhlenbeck process can be interpreted as a <a href="Scaling_limit" class="mw-redirect" title="Scaling limit">scaling limit</a> of a discrete process, in the same way that <a href="Brownian_motion" title="Brownian motion">Brownian motion</a> is a scaling limit of <a href="Random_walks" class="mw-redirect" title="Random walks">random walks</a>. Consider an urn containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> black and white balls. At each step a ball is chosen at random and replaced by a ball of the opposite colour. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{k}}</annotation>

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</math><img src="./33c25229c6989c235f9cbb7908331f6d01d0abfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.013ex; height:2.509ex;" alt="{\displaystyle X_{k}}" loading="lazy"> be the number of black balls in the urn after <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> steps. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {X_{[nt]}-n/2}{\sqrt {n}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {X_{[nt]}-n/2}{\sqrt {n}}}}</annotation>

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</math><img src="./6f0a36af1442653873ecad48c492eb9cb51dee62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.047ex; height:6.843ex;" alt="{\displaystyle {\frac {X_{[nt]}-n/2}{\sqrt {n}}}}" loading="lazy"> converges in law to an Ornstein–Uhlenbeck process as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> tends to infinity. This was obtained by <a href="Mark_Kac" title="Mark Kac">Mark Kac</a>.<a href="#cite_note-FOOTNOTEIglehart1968-11">[11]</a>

Heuristically one may obtain this as follows.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}^{(n)}:={\frac {X_{[nt]}-n/2}{\sqrt {n}}}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{t}^{(n)}:={\frac {X_{[nt]}-n/2}{\sqrt {n}}}}</annotation>

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</math><img src="./603c67a39880a5660851a34035a9a058f96d7b65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.287ex; height:6.843ex;" alt="{\displaystyle X_{t}^{(n)}:={\frac {X_{[nt]}-n/2}{\sqrt {n}}}}" loading="lazy">, and we will obtain the stochastic differential equation at the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }">

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<annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation>

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</math><img src="./a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }" loading="lazy"> limit. First deduce

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<annotation encoding="application/x-tex">{\displaystyle \Delta t=1/n,\quad \Delta X_{t}^{(n)}=X_{t+\Delta t}^{(n)}-X_{t}^{(n)}.}</annotation>

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With this, we can calculate the mean and variance of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta X_{t}^{(n)}}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta X_{t}^{(n)}}</annotation>

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</math><img src="./4f8341ae511a83be8982a7837f5cc9060b6c4416.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.431ex; height:3.676ex;" alt="{\displaystyle \Delta X_{t}^{(n)}}" loading="lazy">, which turns out to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2X_{t}^{(n)}\Delta t}">

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<annotation encoding="application/x-tex">{\displaystyle -2X_{t}^{(n)}\Delta t}</annotation>

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</math><img src="./34072fc1019672ce8d999ba94baaf792129406ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.241ex; height:3.676ex;" alt="{\displaystyle -2X_{t}^{(n)}\Delta t}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}">

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<annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation>

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</math><img src="./8c28867ecd34e2caed12cf38feadf6a81a7ee542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" loading="lazy">. Thus at the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }">

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<annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation>

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</math><img src="./a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }" loading="lazy"> limit, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dX_{t}=-2X_{t}\,dt+dW_{t}}">

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<annotation encoding="application/x-tex">{\displaystyle dX_{t}=-2X_{t}\,dt+dW_{t}}</annotation>

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</math><img src="./1196a004d6b3024ceddb1182fd984831a5707ee9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.304ex; height:2.509ex;" alt="{\displaystyle dX_{t}=-2X_{t}\,dt+dW_{t}}" loading="lazy">, with solution (assuming <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{0}}</annotation>

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</math><img src="./6381fdad2b9f11954b1fc2db08bbaccf634ededa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle X_{0}}" loading="lazy"> distribution is standard normal) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{t}=e^{-2t}W_{e^{4t}}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{t}=e^{-2t}W_{e^{4t}}}</annotation>

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</math><img src="./dd990dee6c96036dbca411c20bc74eb7ba997e94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.364ex; height:3.176ex;" alt="{\displaystyle X_{t}=e^{-2t}W_{e^{4t}}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="In_physics:_noisy_relaxation">In physics: noisy relaxation</h3></div>

The Ornstein–Uhlenbeck process is a prototype of a noisy <a href="Relaxation_(physics)" title="Relaxation (physics)">relaxation process</a>. A canonical example is a <a href="Hooke's_law" title="Hooke's law">Hookean spring</a> (<a href="Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillator</a>) with spring constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> whose dynamics is <a href="Overdamped" class="mw-redirect" title="Overdamped">overdamped</a>

with friction coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

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</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy">. In the presence of thermal fluctuations with <a href="Temperature" title="Temperature">temperature</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">, the length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}">

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<annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation>

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</math><img src="./d54c275db3a1e620737b58e143b0818107fa5f5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" loading="lazy"> of the spring fluctuates around the spring rest length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

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</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy">; its stochastic dynamics is described by an Ornstein–Uhlenbeck process with

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\theta &=k/\gamma ,\\\mu &=x_{0},\\\sigma &={\sqrt {2k_{B}T/\gamma }},\end{aligned}}}</annotation>

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</math><img src="./9c60b87bb429fe5cf1cdbd58fcced2826e73f7e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:16.137ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}\theta &=k/\gamma ,\\\mu &=x_{0},\\\sigma &={\sqrt {2k_{B}T/\gamma }},\end{aligned}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> is derived from the <a href="Stokes%E2%80%93Einstein_equation" class="mw-redirect" title="Stokes–Einstein equation">Stokes–Einstein equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=\sigma ^{2}/2=k_{B}T/\gamma }">

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<annotation encoding="application/x-tex">{\displaystyle D=\sigma ^{2}/2=k_{B}T/\gamma }</annotation>

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</math><img src="./fe6c33ead97fc3d15eb99dd6eb97e88f4241ed37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.583ex; height:3.176ex;" alt="{\displaystyle D=\sigma ^{2}/2=k_{B}T/\gamma }" loading="lazy"> for the effective diffusion constant.<a href="#cite_note-FOOTNOTENørrelykkeFlyvbjerg2011-12">[12]</a><a href="#cite_note-FOOTNOTEGoerlichLiAlbertManfredi2021-13">[13]</a> This model has been used to characterize the motion of a Brownian particle in an <a href="Optical_trap" class="mw-redirect" title="Optical trap">optical trap</a>.<a href="#cite_note-FOOTNOTEGoerlichLiAlbertManfredi2021-13">[13]</a><a href="#cite_note-FOOTNOTELiSentissiAzziniSchnoering2019-14">[14]</a>

At equilibrium, the spring stores an average energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle E\rangle =k\langle (x-x_{0})^{2}\rangle /2=k_{B}T/2}">

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<annotation encoding="application/x-tex">{\displaystyle \langle E\rangle =k\langle (x-x_{0})^{2}\rangle /2=k_{B}T/2}</annotation>

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</math><img src="./d53e1f8167446af79774f704bb512f6f907abb88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.197ex; height:3.176ex;" alt="{\displaystyle \langle E\rangle =k\langle (x-x_{0})^{2}\rangle /2=k_{B}T/2}" loading="lazy"> in accordance with the <a href="Equipartition_theorem" title="Equipartition theorem">equipartition theorem</a>.<a href="#cite_note-FOOTNOTENelson1967-15">[15]</a>

<div class="mw-heading mw-heading3"><h3 id="In_financial_mathematics">In financial mathematics</h3></div>

The Ornstein–Uhlenbeck process is used in the <a href="Vasicek_model" title="Vasicek model">Vasicek model</a> of the interest rate.<a href="#cite_note-FOOTNOTEBjörk2009375,_381-16">[16]</a> The Ornstein–Uhlenbeck process is one of several approaches used to model (with modifications) interest rates, currency <a href="Exchange_rate" title="Exchange rate">exchange rates</a>, and commodity prices stochastically. The parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> represents the equilibrium or mean value supported by <a href="Fundamental_analysis" title="Fundamental analysis">fundamentals</a>; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

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</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> the degree of <a href="Volatility_(finance)" title="Volatility (finance)">volatility</a> around it caused by <a href="Shock_(economics)" title="Shock (economics)">shocks</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

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<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

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</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> 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 <a href="Pairs_trade" title="Pairs trade">pairs trade</a>.<a href="#cite_note-FOOTNOTELeungLi2016-17">[17]</a><a href="#cite_note-18">[18]</a><a href="#cite_note-19">[19]</a>

A further implementation of the Ornstein–Uhlenbeck process is derived by Marcello Minenna in order to model the <a href="Stock" title="Stock">stock</a> return under a <a href="Lognormal_distribution" class="mw-redirect" title="Lognormal distribution">lognormal distribution</a> dynamics. This modeling aims at the determination of <a href="Confidence_interval" title="Confidence interval">confidence interval</a> to predict <a href="Market_abuse" title="Market abuse">market abuse</a> phenomena.<a href="#cite_note-20">[20]</a><a href="#cite_note-21">[21]</a>

<div class="mw-heading mw-heading3"><h3 id="In_evolutionary_biology">In evolutionary biology</h3></div>

The Ornstein–Uhlenbeck process has been proposed as an improvement over a Brownian motion model for modeling the change in organismal <a href="Phenotypes" class="mw-redirect" title="Phenotypes">phenotypes</a> over time.<a href="#cite_note-FOOTNOTEMartins1994193–209-22">[22]</a> A Brownian motion model implies that the phenotype can move without limit, whereas for most phenotypes natural selection imposes a cost for moving too far in either direction. A meta-analysis of 250 fossil phenotype time-series showed that an Ornstein–Uhlenbeck model was the best fit for 115 (46%) of the examined time series, supporting stasis as a common evolutionary pattern.<a href="#cite_note-FOOTNOTEHunt2007-23">[23]</a> This said, there are certain challenges to its use: model selection mechanisms are often biased towards preferring an OU process without sufficient support, and misinterpretation is easy to the unsuspecting data scientist.<a href="#cite_note-FOOTNOTECornuault2022-24">[24]</a>

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

It is possible to define a Lévy-driven Ornstein–Uhlenbeck process, in which the background driving process is a <a href="L%C3%A9vy_process" title="Lévy process">Lévy process</a> instead of a Wiener process:<a href="#cite_note-FOOTNOTEJespersenMetzlerFogedby1999-25">[25]</a><a href="#cite_note-FOOTNOTEFinkKlüppelberg2011-26">[26]</a>

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<annotation encoding="application/x-tex">{\displaystyle dx_{t}=-\theta \,x_{t}\,dt+\sigma \,dL_{t}}</annotation>

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</math><img src="./9cfc9d4e599514741fcb44532501ee78dcdb29ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.536ex; height:2.509ex;" alt="{\displaystyle dx_{t}=-\theta \,x_{t}\,dt+\sigma \,dL_{t}}" loading="lazy"></dd></dl>

Here, the differential of the Wiener process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{t}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{t}}</annotation>

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</math><img src="./50680c5535c83badfa630dba63b583d2eeaa2977.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.02ex; height:2.509ex;" alt="{\displaystyle W_{t}}" loading="lazy"> has been replaced with the differential of a Lévy process <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{t}}">

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<annotation encoding="application/x-tex">{\displaystyle L_{t}}</annotation>

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</math><img src="./8e352de9d624cc3457a63670bcd92f2d2f8467a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle L_{t}}" loading="lazy">.

In addition, in finance, stochastic processes are used where the volatility increases for larger values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. In particular, the <a href="CKLS_process" class="mw-redirect" title="CKLS process">CKLS process</a> (Chan–Karolyi–Longstaff–Sanders)<a href="#cite_note-FOOTNOTEChanKarolyiLongstaffSanders1992-27">[27]</a> with the volatility term replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \,x^{\gamma }\,dW_{t}}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma \,x^{\gamma }\,dW_{t}}</annotation>

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</math><img src="./75e6a0809ed2be25fae6c63f7a202bd0a5640ed7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.794ex; height:2.676ex;" alt="{\displaystyle \sigma \,x^{\gamma }\,dW_{t}}" loading="lazy"> can be solved in closed form for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =1}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma =1}</annotation>

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</math><img src="./5682ebb86d6f024a15f4a2c1c7cb08412720bcaf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.676ex;" alt="{\displaystyle \gamma =1}" loading="lazy">, as well as for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =0}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma =0}</annotation>

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</math><img src="./0a5e84cac32e896a80a89f8cd1917c2defcf4108.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.523ex; height:2.676ex;" alt="{\displaystyle \gamma =0}" loading="lazy">, which corresponds to the conventional OU process. Another special case is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =1/2}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma =1/2}</annotation>

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</math><img src="./bd9b55bc843569ad5b00d7bb7d3010394dd58b6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.848ex; height:2.843ex;" alt="{\displaystyle \gamma =1/2}" loading="lazy">, which corresponds to the <a href="Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross model</a> (CIR-model).

<div class="mw-heading mw-heading3"><h3 id="Higher_dimensions">Higher dimensions</h3></div>

A multi-dimensional version of the Ornstein–Uhlenbeck process, denoted by the N-dimensional vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} _{t}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{t}}</annotation>

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</math><img src="./4e2853a10087b64e68ecf97f07e797b844b70c3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.237ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{t}}" loading="lazy">, can be defined from

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<annotation encoding="application/x-tex">{\displaystyle d\mathbf {x} _{t}=-{\boldsymbol {\beta }}\,\mathbf {x} _{t}\,dt+{\boldsymbol {\sigma }}\,d\mathbf {W} _{t}.}</annotation>

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</math><img src="./337a37572f85f8a588c0da8a8456fd08aadad471.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.233ex; height:2.509ex;" alt="{\displaystyle d\mathbf {x} _{t}=-{\boldsymbol {\beta }}\,\mathbf {x} _{t}\,dt+{\boldsymbol {\sigma }}\,d\mathbf {W} _{t}.}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {W} _{t}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {W} _{t}}</annotation>

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</math><img src="./2e9e6082984a94fa7c286ea644bc47a1bbb271fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.589ex; height:2.509ex;" alt="{\displaystyle \mathbf {W} _{t}}" loading="lazy"> is an N-dimensional Wiener process, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}}">

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</math><img src="./702cafc420cc00c54896f6d125112820956aaf6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.534ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}}</annotation>

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</math><img src="./e45fe1b9d8dcbc3103fc7805d69798bfe5ca5b16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.594ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\sigma }}}" loading="lazy"> are constant N×N matrices.<a href="#cite_note-FOOTNOTEGardiner1985109-28">[28]</a> The solution is

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{t}=e^{-{\boldsymbol {\beta }}t}\mathbf {x} _{0}+\int _{0}^{t}e^{-{\boldsymbol {\beta }}(t-t')}{\boldsymbol {\sigma }}\,d\mathbf {W} _{t'}}</annotation>

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</math><img src="./0ed7208e4bba3b115e62d58b5b17bd4c6a7177e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.865ex; height:6.176ex;" alt="{\displaystyle \mathbf {x} _{t}=e^{-{\boldsymbol {\beta }}t}\mathbf {x} _{0}+\int _{0}^{t}e^{-{\boldsymbol {\beta }}(t-t')}{\boldsymbol {\sigma }}\,d\mathbf {W} _{t'}}" loading="lazy"></dd></dl>

and the mean is

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {E} (\mathbf {x} _{t})=e^{-{\boldsymbol {\beta }}t}\operatorname {E} (\mathbf {x} _{0}).}</annotation>

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</math><img src="./2fe52590988608017808e98aac763e3b8f6506cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.892ex; height:3.176ex;" alt="{\displaystyle \operatorname {E} (\mathbf {x} _{t})=e^{-{\boldsymbol {\beta }}t}\operatorname {E} (\mathbf {x} _{0}).}" loading="lazy"></dd></dl>

These expressions make use of the <a href="Matrix_exponential" title="Matrix exponential">matrix exponential</a>.

The process can also be described in terms of the probability density function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\mathbf {x} ,t)}">

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<annotation encoding="application/x-tex">{\displaystyle P(\mathbf {x} ,t)}</annotation>

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</math><img src="./ae1abfafb8269770ffd56b37048ee421a8682e28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.839ex; height:2.843ex;" alt="{\displaystyle P(\mathbf {x} ,t)}" loading="lazy">, which satisfies the Fokker–Planck equation<a href="#cite_note-FOOTNOTEGardiner198597-29">[29]</a>

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</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial P}{\partial t}}=\sum _{i,j}\beta _{ij}{\frac {\partial }{\partial x_{i}}}(x_{j}P)+\sum _{i,j}D_{ij}{\frac {\partial ^{2}P}{\partial x_{i}\,\partial x_{j}}},}</annotation>

</semantics>

</math><img src="./1f6230f79796a27d0ceb679f8b8de46fffb9f477.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:42.47ex; height:7.176ex;" alt="{\displaystyle {\frac {\partial P}{\partial t}}=\sum _{i,j}\beta _{ij}{\frac {\partial }{\partial x_{i}}}(x_{j}P)+\sum _{i,j}D_{ij}{\frac {\partial ^{2}P}{\partial x_{i}\,\partial x_{j}}},}" loading="lazy"></dd></dl>

where the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {D}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {D}}}</annotation>

</semantics>

</math><img src="./ff7154d73f1915f70bc06893818ef63386fb0d36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.18ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {D}}}" loading="lazy"> with components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{ij}}</annotation>

</semantics>

</math><img src="./88badd2360f6d55865a1e8f1b4e3994451bcc075.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.401ex; height:2.843ex;" alt="{\displaystyle D_{ij}}" loading="lazy"> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {D}}={\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{T}/2}">

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {D}}={\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{T}/2}</annotation>

</semantics>

</math><img src="./865d467f15596e46b2b3ac1219acd6e931ee63f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.181ex; height:3.176ex;" alt="{\displaystyle {\boldsymbol {D}}={\boldsymbol {\sigma }}{\boldsymbol {\sigma }}^{T}/2}" loading="lazy">. Like for the one-dimensional case, the process is a linear transformation of Gaussian random variables, and therefore itself must be Gaussian. Because of this, the transition probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\mathbf {x} ,t\mid \mathbf {x} ',t')}">

</mrow>

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P(\mathbf {x} ,t\mid \mathbf {x} ',t')}</annotation>

</semantics>

</math><img src="./0c43487f02219875d96eaf870f9d7e4884562532.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.43ex; height:3.009ex;" alt="{\displaystyle P(\mathbf {x} ,t\mid \mathbf {x} ',t')}" loading="lazy"> is a Gaussian which can be written down explicitly. If the real parts of the eigenvalues of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}}</annotation>

</semantics>

</math><img src="./702cafc420cc00c54896f6d125112820956aaf6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.534ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\beta }}}" loading="lazy"> are larger than zero, a stationary solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\text{st}}(\mathbf {x} )}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\text{st}}(\mathbf {x} )}</annotation>

</semantics>

</math><img src="./ad00dd41a0cd58d65a4306a2d7dd8e9c627434b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.232ex; height:2.843ex;" alt="{\displaystyle P_{\text{st}}(\mathbf {x} )}" loading="lazy"> moreover exists, given by

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\text{st}}(\mathbf {x} )=(2\pi )^{-N/2}(\det {\boldsymbol {\omega }})^{-1/2}\exp \left(-{\frac {1}{2}}\mathbf {x} ^{T}{\boldsymbol {\omega }}^{-1}\mathbf {x} \right),}</annotation>

</semantics>

</math><img src="./56636a9f98afeb8c151f7695887c651eb11ad02c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.347ex; height:6.176ex;" alt="{\displaystyle P_{\text{st}}(\mathbf {x} )=(2\pi )^{-N/2}(\det {\boldsymbol {\omega }})^{-1/2}\exp \left(-{\frac {1}{2}}\mathbf {x} ^{T}{\boldsymbol {\omega }}^{-1}\mathbf {x} \right),}" loading="lazy"></dd></dl>

where the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\omega }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}}</annotation>

</semantics>

</math><img src="./7cb8af7a2f64af348e559652b6b1f0d2415ba444.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.669ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\omega }}}" loading="lazy"> is determined from the <a href="Lyapunov_equation" title="Lyapunov equation">Lyapunov equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\beta }}{\boldsymbol {\omega }}+{\boldsymbol {\omega }}{\boldsymbol {\beta }}^{T}=2{\boldsymbol {D}}}">

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\beta }}{\boldsymbol {\omega }}+{\boldsymbol {\omega }}{\boldsymbol {\beta }}^{T}=2{\boldsymbol {D}}}</annotation>

</semantics>

</math><img src="./ed69569e0c7a6eb456629dccbd0472f1ce5442bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.076ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {\beta }}{\boldsymbol {\omega }}+{\boldsymbol {\omega }}{\boldsymbol {\beta }}^{T}=2{\boldsymbol {D}}}" loading="lazy">.<a href="#cite_note-FOOTNOTERisken1989-5">[5]</a>

<ul><li><a href="Stochastic_calculus" title="Stochastic calculus">Stochastic calculus</a></li>

<li><a href="Wiener_process" title="Wiener process">Wiener process</a></li>

<li><a href="Gaussian_process" title="Gaussian process">Gaussian process</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li>The <a href="Vasicek_model" title="Vasicek model">Vasicek model</a> of <a href="Interest_rates" class="mw-redirect" title="Interest rates">interest rates</a></li>

<li><a href="Short-rate_model" title="Short-rate model">Short-rate model</a></li>

<li><a href="Diffusion" title="Diffusion">Diffusion</a></li>

<li><a href="Fluctuation-dissipation_theorem" class="mw-redirect" title="Fluctuation-dissipation theorem">Fluctuation-dissipation theorem</a></li>

<li><a href="Klein%E2%80%93Kramers_equation" title="Klein–Kramers equation">Klein–Kramers equation</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist">

<li id="cite_note-FOOTNOTEDoob1942-1">^ <a href="#cite_ref-FOOTNOTEDoob1942_1-0">a</a> <a href="#cite_ref-FOOTNOTEDoob1942_1-1">b</a> <a href="#CITEREFDoob1942">Doob 1942</a>.

</li>

<li id="cite_note-FOOTNOTEKaratzasShreve1991358-2"><a href="#cite_ref-FOOTNOTEKaratzasShreve1991358_2-0">^</a> <a href="#CITEREFKaratzasShreve1991">Karatzas & Shreve 1991</a>, p. 358.

</li>

<li id="cite_note-FOOTNOTEGard1988115-3"><a href="#cite_ref-FOOTNOTEGard1988115_3-0">^</a> <a href="#CITEREFGard1988">Gard 1988</a>, p. 115.

</li>

<li id="cite_note-FOOTNOTEGardiner1985-4"><a href="#cite_ref-FOOTNOTEGardiner1985_4-0">^</a> <a href="#CITEREFGardiner1985">Gardiner 1985</a>.

</li>

<li id="cite_note-FOOTNOTERisken1989-5">^ <a href="#cite_ref-FOOTNOTERisken1989_5-0">a</a> <a href="#cite_ref-FOOTNOTERisken1989_5-1">b</a> <a href="#cite_ref-FOOTNOTERisken1989_5-2">c</a> <a href="#CITEREFRisken1989">Risken 1989</a>.

</li>

<li id="cite_note-FOOTNOTELawler2006-6"><a href="#cite_ref-FOOTNOTELawler2006_6-0">^</a> <a href="#CITEREFLawler2006">Lawler 2006</a>.

</li>

<li id="cite_note-FOOTNOTEGardiner1985106-7"><a href="#cite_ref-FOOTNOTEGardiner1985106_7-0">^</a> <a href="#CITEREFGardiner1985">Gardiner 1985</a>, p. 106.

</li>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFHolmes-Cerfon2022" class="citation web cs1">Holmes-Cerfon, Miranda (2022). <a rel="nofollow" class="external text" href="https://cims.nyu.edu/~holmes/teaching/asa22/handout-Lecture12_2022.pdf">"Lecture 12: Detailed balance and Eigenfunction methods"</a> (PDF).</cite>

</li>

<li id="cite_note-FOOTNOTEAït-Sahalia2002223–262-9"><a href="#cite_ref-FOOTNOTEAït-Sahalia2002223–262_9-0">^</a> <a href="#CITEREFAït-Sahalia2002">Aït-Sahalia 2002</a>, pp. 223–262.

</li>

<li id="cite_note-FOOTNOTEKloedenPlatenSchurz1994-10"><a href="#cite_ref-FOOTNOTEKloedenPlatenSchurz1994_10-0">^</a> <a href="#CITEREFKloedenPlatenSchurz1994">Kloeden, Platen & Schurz 1994</a>.

</li>

<li id="cite_note-FOOTNOTEIglehart1968-11"><a href="#cite_ref-FOOTNOTEIglehart1968_11-0">^</a> <a href="#CITEREFIglehart1968">Iglehart 1968</a>.

</li>

<li id="cite_note-FOOTNOTENørrelykkeFlyvbjerg2011-12"><a href="#cite_ref-FOOTNOTENørrelykkeFlyvbjerg2011_12-0">^</a> <a href="#CITEREFNørrelykkeFlyvbjerg2011">Nørrelykke & Flyvbjerg 2011</a>.

</li>

<li id="cite_note-FOOTNOTEGoerlichLiAlbertManfredi2021-13">^ <a href="#cite_ref-FOOTNOTEGoerlichLiAlbertManfredi2021_13-0">a</a> <a href="#cite_ref-FOOTNOTEGoerlichLiAlbertManfredi2021_13-1">b</a> <a href="#CITEREFGoerlichLiAlbertManfredi2021">Goerlich et al. 2021</a>.

</li>

<li id="cite_note-FOOTNOTELiSentissiAzziniSchnoering2019-14"><a href="#cite_ref-FOOTNOTELiSentissiAzziniSchnoering2019_14-0">^</a> <a href="#CITEREFLiSentissiAzziniSchnoering2019">Li et al. 2019</a>.

</li>

<li id="cite_note-FOOTNOTENelson1967-15"><a href="#cite_ref-FOOTNOTENelson1967_15-0">^</a> <a href="#CITEREFNelson1967">Nelson 1967</a>.

</li>

<li id="cite_note-FOOTNOTEBjörk2009375,_381-16"><a href="#cite_ref-FOOTNOTEBjörk2009375,_381_16-0">^</a> <a href="#CITEREFBjörk2009">Björk 2009</a>, pp. 375, 381.

</li>

<li id="cite_note-FOOTNOTELeungLi2016-17"><a href="#cite_ref-FOOTNOTELeungLi2016_17-0">^</a> <a href="#CITEREFLeungLi2016">Leung & Li 2016</a>.

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <a rel="nofollow" class="external text" href="http://www.cs.sunysb.edu/~skiena/691/lectures/lecture23.pdf">Advantages of Pair Trading: Market Neutrality</a>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <a rel="nofollow" class="external text" href="http://www.ms.unimelb.edu.au/publications/RampertshammerStefan.pdf">An Ornstein–Uhlenbeck Framework for Pairs Trading</a>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.risk.net/regulation/1528679/detecting-market-abuse">"Detecting Market Abuse"</a>. Risk Magazine. 2 November 2004.</cite>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.consob.it/documents/1912911/2006254/qdf54en.pdf/d31b160c-9ba5-e08d-d39d-a78bddfc698e">"The detection of Market Abuse on financial markets: a quantitative approach"</a>. Consob – The Italian Securities and Exchange Commission.</cite>

</li>

<li id="cite_note-FOOTNOTEMartins1994193–209-22"><a href="#cite_ref-FOOTNOTEMartins1994193–209_22-0">^</a> <a href="#CITEREFMartins1994">Martins 1994</a>, pp. 193–209.

</li>

<li id="cite_note-FOOTNOTEHunt2007-23"><a href="#cite_ref-FOOTNOTEHunt2007_23-0">^</a> <a href="#CITEREFHunt2007">Hunt 2007</a>.

</li>

<li id="cite_note-FOOTNOTECornuault2022-24"><a href="#cite_ref-FOOTNOTECornuault2022_24-0">^</a> <a href="#CITEREFCornuault2022">Cornuault 2022</a>.

</li>

<li id="cite_note-FOOTNOTEJespersenMetzlerFogedby1999-25"><a href="#cite_ref-FOOTNOTEJespersenMetzlerFogedby1999_25-0">^</a> <a href="#CITEREFJespersenMetzlerFogedby1999">Jespersen, Metzler & Fogedby 1999</a>.

</li>

<li id="cite_note-FOOTNOTEFinkKlüppelberg2011-26"><a href="#cite_ref-FOOTNOTEFinkKlüppelberg2011_26-0">^</a> <a href="#CITEREFFinkKlüppelberg2011">Fink & Klüppelberg 2011</a>.

</li>

<li id="cite_note-FOOTNOTEChanKarolyiLongstaffSanders1992-27"><a href="#cite_ref-FOOTNOTEChanKarolyiLongstaffSanders1992_27-0">^</a> <a href="#CITEREFChanKarolyiLongstaffSanders1992">Chan et al. 1992</a>.

</li>

<li id="cite_note-FOOTNOTEGardiner1985109-28"><a href="#cite_ref-FOOTNOTEGardiner1985109_28-0">^</a> <a href="#CITEREFGardiner1985">Gardiner 1985</a>, p. 109.

</li>

<li id="cite_note-FOOTNOTEGardiner198597-29"><a href="#cite_ref-FOOTNOTEGardiner198597_29-0">^</a> <a href="#CITEREFGardiner1985">Gardiner 1985</a>, p. 97.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<ul><li><cite id="CITEREFAït-Sahalia2002" class="citation journal cs1"><a href="Yacine_A%C3%AFt-Sahalia" title="Yacine Aït-Sahalia">Aït-Sahalia, Y.</a> (April 2002). "Maximum Likelihood Estimation of Discretely Sampled Diffusion: A Closed-Form Approximation Approach". <a href="Econometrica" title="Econometrica">Econometrica</a>. 70 (1): 223–262. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2F1468-0262.00274">10.1111/1468-0262.00274</a>.</cite></li>

<li><cite id="CITEREFBibbonaPanfiloTavella2008" class="citation journal cs1">Bibbona, E.; Panfilo, G.; Tavella, P. (2008). "The Ornstein–Uhlenbeck process as a model of a low pass filtered white noise". Metrologia. 45 (6): S117 – S126. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008Metro..45S.117B">2008Metro..45S.117B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0026-1394%2F45%2F6%2FS17">10.1088/0026-1394/45/6/S17</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2318%2F58227">2318/58227</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:56160285">56160285</a>.</cite></li>

<li><cite id="CITEREFBjörk2009" class="citation book cs1">Björk, Tomas (2009). Arbitrage Theory in Continuous Time (3rd ed.). Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-957474-2</bdi>.</cite></li>

<li><cite id="CITEREFChanKarolyiLongstaffSanders1992" class="citation journal cs1">Chan, K. C.; Karolyi, G. A.; Longstaff, F. A.; Sanders, A. B. (1992). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1540-6261.1992.tb04011.x">"An empirical comparison of alternative models of the short-term interest rate"</a>. <a href="Journal_of_Finance" class="mw-redirect" title="Journal of Finance">Journal of Finance</a>. 47 (3): 1209–1227. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1540-6261.1992.tb04011.x">10.1111/j.1540-6261.1992.tb04011.x</a>.</cite></li>

<li><cite id="CITEREFCornuault2022" class="citation journal cs1">Cornuault, Josselin (2022). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9558839">"Bayesian Analyses of Comparative Data with the Ornstein–Uhlenbeck Model: Potential Pitfalls"</a>. Systematic Biology. 71 (6): 1524–1540. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fsysbio%2Fsyac036">10.1093/sysbio/syac036</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9558839">9558839</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/35583306">35583306</a>.</cite></li>

<li><cite id="CITEREFDoob1942" class="citation journal cs1"><a href="Joseph_Leo_Doob" class="mw-redirect" title="Joseph Leo Doob">Doob, J.L.</a> (April 1942). "The Brownian Movement and Stochastic Equations". <a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>. 43 (2): 351–369. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1968873">10.2307/1968873</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1968873">1968873</a>.</cite></li>

<li><cite id="CITEREFFinkKlüppelberg2011" class="citation journal cs1">Fink, Holger; Klüppelberg, Claudia (2011-02-01). "Fractional Lévy-driven Ornstein–Uhlenbeck processes and stochastic differential equations". Bernoulli. 17 (1). <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1102.1830">1102.1830</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3150%2F10-bej281">10.3150/10-bej281</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1350-7265">1350-7265</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:9269536">9269536</a>.</cite></li>

<li><cite id="CITEREFHunt2007" class="citation journal cs1">Hunt, G. (2007-11-14). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2141789">"The relative importance of directional change, random walks, and stasis in the evolution of fossil lineages"</a>. Proceedings of the National Academy of Sciences. 104 (47): 18404–18408. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.0704088104">10.1073/pnas.0704088104</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0027-8424">0027-8424</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2141789">2141789</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18003931">18003931</a>.</cite></li>

<li><cite id="CITEREFGard1988" class="citation cs2">Gard, Thomas C. (1988), Introduction to Stochastic Differential Equations, Marcel Dekker, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8247-7776-0</bdi></cite></li>

<li><cite id="CITEREFGardiner1985" class="citation cs2"><a href="Crispin_Gardiner" title="Crispin Gardiner">Gardiner, Crispin W.</a> (1985), Handbook of Stochastic Methods (2nd ed.), Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-15607-1</bdi></cite></li>

<li><cite id="CITEREFGillespie1996" class="citation journal cs1">Gillespie, D. T. (1996). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1233795">"Exact numerical simulation of the Ornstein–Uhlenbeck process and its integral"</a>. <a href="Physical_Review" title="Physical Review">Phys. Rev. E</a>. 54 (2): 2084–2091. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1996PhRvE..54.2084G">1996PhRvE..54.2084G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevE.54.2084">10.1103/PhysRevE.54.2084</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/9965289">9965289</a>.</cite></li>

<li><cite id="CITEREFIglehart1968" class="citation journal cs1">Iglehart, Donald L. (June 1968). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177698318">"Limit Theorems for the Multi-urn Ehrenfest Model"</a>. The Annals of Mathematical Statistics. 39 (3): 864–876. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faoms%2F1177698318">10.1214/aoms/1177698318</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-4851">0003-4851</a>.</cite></li>

<li><cite id="CITEREFKaratzasShreve1991" class="citation cs2">Karatzas, Ioannis; Shreve, Steven E. (1991), Brownian Motion and Stochastic Calculus (2nd ed.), Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-97655-6</bdi></cite></li>

<li><cite id="CITEREFGoerlichLiAlbertManfredi2021" class="citation journal cs1">Goerlich, Rémi; Li, Minghao; Albert, Samuel; Manfredi, Giovanni; Hervieux, Paul-Antoine; Genet, Cyriaque (2021-03-19). "Noise and ergodic properties of Brownian motion in an optical tweezer: Looking at regime crossovers in an Ornstein-Uhlenbeck process". Physical Review E. 103 (3): 032132. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2007.12246">2007.12246</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2021PhRvE.103c2132G">2021PhRvE.103c2132G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysreve.103.032132">10.1103/physreve.103.032132</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2470-0045">2470-0045</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/33862817">33862817</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:220768666">220768666</a>.</cite></li>

<li><cite id="CITEREFJespersenMetzlerFogedby1999" class="citation journal cs1">Jespersen, Sune; Metzler, Ralf; Fogedby, Hans C. (1999-03-01). "Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions". Physical Review E. 59 (3): 2736–2745. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/9810176">cond-mat/9810176</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999PhRvE..59.2736J">1999PhRvE..59.2736J</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysreve.59.2736">10.1103/physreve.59.2736</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1063-651X">1063-651X</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:51944991">51944991</a>.</cite></li>

<li><cite id="CITEREFKloedenPlatenSchurz1994" class="citation book cs1">Kloeden, Peter E.; Platen, Eckhard; Schurz, Henri (1994). Numerical solution of SDE through computer experiments. Berlin: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-57074-8</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/29788831">29788831</a>.</cite></li>

<li><cite id="CITEREFLawler2006" class="citation book cs1"><a href="Greg_Lawler" title="Greg Lawler">Lawler, Gregory F.</a> (2006). Introduction to Stochastic Processes (2nd ed.). Chapman & Hall/CRC. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1584886518</bdi>.</cite></li>

<li><cite id="CITEREFLeungLi2016" class="citation book cs1">Leung, Tim; Li, Xin (2016). Optimal Mean-Reversion Trading: Mathematical Analysis and Practical Applications. <a href="World_Scientific_Publishing_Co." class="mw-redirect" title="World Scientific Publishing Co.">World Scientific Publishing Co.</a> <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-9814725910</bdi>.</cite></li>

<li><cite id="CITEREFLeungLi2015" class="citation journal cs1">Leung, Tim; Li, Xin (2015). "Optimal Mean Reversion Trading with Transaction Costs and Stop-Loss Exit". International Journal of Theoretical and Applied Finance. 18 (3): 1550020. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1411.5062">1411.5062</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS021902491550020X">10.1142/S021902491550020X</a>.</cite></li>

<li><cite id="CITEREFLiSentissiAzziniSchnoering2019" class="citation journal cs1">Li, Minghao; Sentissi, Oussama; Azzini, Stefano; Schnoering, Gabriel; Canaguier-Durand, Antoine; Genet, Cyriaque (2019-12-10). "Subfemtonewton force fields measured with ergodic Brownian ensembles". Physical Review A. 100 (6): 063816. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1908.00610">1908.00610</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019PhRvA.100f3816L">2019PhRvA.100f3816L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysreva.100.063816">10.1103/physreva.100.063816</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2469-9926">2469-9926</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:199405409">199405409</a>.</cite></li>

<li><cite id="CITEREFMartins1994" class="citation journal cs1">Martins, Emilia P. (1994). "Estimating the Rate of Phenotypic Evolution from Comparative Data". <a href="The_American_Naturalist" title="The American Naturalist">The American Naturalist</a>. 144 (2): 193–209. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994ANat..144..193M">1994ANat..144..193M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F285670">10.1086/285670</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-0147">0003-0147</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:85300707">85300707</a>.</cite></li>

<li><cite id="CITEREFNørrelykkeFlyvbjerg2011" class="citation journal cs1">Nørrelykke, Simon F.; Flyvbjerg, Henrik (2011-04-04). "Harmonic oscillator in heat bath: Exact simulation of time-lapse-recorded data and exact analytical benchmark statistics". Physical Review E. 83 (4): 041103. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1102.0524">1102.0524</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011PhRvE..83d1103N">2011PhRvE..83d1103N</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2Fphysreve.83.041103">10.1103/physreve.83.041103</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1539-3755">1539-3755</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21599111">21599111</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18518657">18518657</a>.</cite></li>

<li><cite id="CITEREFRisken1989" class="citation book cs1">Risken, H. (1989). The Fokker–Planck Equation: Methods of Solution and Applications. New York: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0387504988</bdi>.</cite></li>

<li><cite id="CITEREFNelson1967" class="citation book cs1">Nelson, Edward (1967). <a rel="nofollow" class="external text" href="http://web.math.princeton.edu/~nelson/books/bmotion.pdf">Dynamical theories of Brownian motion</a> (PDF). Princeton, N.J.: Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-691-07950-1</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/769464">769464</a>.</cite></li>

<li><cite id="CITEREFUhlenbeckOrnstein1930" class="citation journal cs1">Uhlenbeck, G. E.; Ornstein, L. S. (1930). "On the theory of Brownian Motion". Phys. Rev. 36 (5): 823–841. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1930PhRv...36..823U">1930PhRv...36..823U</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.36.823">10.1103/PhysRev.36.823</a>.</cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://ssrn.com/abstract=1109160">A Stochastic Processes Toolkit for Risk Management</a>, Damiano Brigo, Antonio Dalessandro, Matthias Neugebauer and Fares Triki</li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150619164944/http://www.sitmo.com/article/calibrating-the-ornstein-uhlenbeck-model/">Simulating and Calibrating the Ornstein–Uhlenbeck process</a>, M. A. van den Berg</li>

<li><a rel="nofollow" class="external text" href="http://www.investmentscience.com/Content/howtoArticles/MLE_for_OR_mean_reverting.pdf">Maximum likelihood estimation of mean reverting processes</a>, Jose Carlos Garcia Franco</li>

<li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150920231636/http://turingfinance.com/interactive-stochastic-processes/">"Interactive Web Application: Stochastic Processes used in Quantitative Finance"</a>. Archived from <a rel="nofollow" class="external text" href="http://turingfinance.com/interactive-stochastic-processes/">the original</a> on 2015-09-20. Retrieved 2015-07-03.</cite></li></ul>

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</style><div id="Stochastic_processes496" style="font-size:114%;margin:0 4em"><a href="Stochastic_process" title="Stochastic process">Stochastic processes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Discrete-time_stochastic_process" class="mw-redirect" title="Discrete-time stochastic process">Discrete time</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bernoulli_process" title="Bernoulli process">Bernoulli process</a></li>

<li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Chinese_restaurant_process" title="Chinese restaurant process">Chinese restaurant process</a></li>

<li><a href="Galton%E2%80%93Watson_process" title="Galton–Watson process">Galton–Watson process</a></li>

<li><a href="Independent_and_identically_distributed_random_variables" title="Independent and identically distributed random variables">Independent and identically distributed random variables</a></li>

<li><a href="Markov_chain" title="Markov chain">Markov chain</a></li>

<li><a href="Moran_process" title="Moran process">Moran process</a></li>

<li><a href="Random_walk" title="Random walk">Random walk</a>

<ul><li><a href="Loop-erased_random_walk" title="Loop-erased random walk">Loop-erased</a></li>

<li><a href="Self-avoiding_walk" title="Self-avoiding walk">Self-avoiding</a></li>

<li><a href="Biased_random_walk_on_a_graph" title="Biased random walk on a graph"> Biased</a></li>

<li><a href="Maximal_entropy_random_walk" title="Maximal entropy random walk">Maximal entropy</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Continuous-time_stochastic_process" title="Continuous-time stochastic process">Continuous time</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Additive_process" title="Additive process">Additive process</a></li>

<li><a href="Airy_process" title="Airy process">Airy process</a></li>

<li><a href="Bessel_process" title="Bessel process">Bessel process</a></li>

<li><a href="Birth%E2%80%93death_process" title="Birth–death process">Birth–death process</a>

<ul><li><a href="Birth_process" title="Birth process">pure birth</a></li></ul></li>

<li><a href="Wiener_process" title="Wiener process">Brownian motion</a>

<ul><li><a href="Brownian_bridge" title="Brownian bridge">Bridge</a></li>

<li><a href="Dyson_Brownian_motion" title="Dyson Brownian motion">Dyson</a></li>

<li><a href="Brownian_excursion" title="Brownian excursion">Excursion</a></li>

<li><a href="Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional</a></li>

<li><a href="Geometric_Brownian_motion" title="Geometric Brownian motion">Geometric</a></li>

<li><a href="Brownian_meander" title="Brownian meander">Meander</a></li></ul></li>

<li><a href="Cauchy_process" title="Cauchy process">Cauchy process</a></li>

<li><a href="Contact_process_(mathematics)" title="Contact process (mathematics)">Contact process</a></li>

<li><a href="Continuous-time_random_walk" title="Continuous-time random walk">Continuous-time random walk</a></li>

<li><a href="Cox_process" title="Cox process">Cox process</a></li>

<li><a href="Diffusion_process" title="Diffusion process">Diffusion process</a></li>

<li><a href="Empirical_process" title="Empirical process">Empirical process</a></li>

<li><a href="Feller_process" title="Feller process">Feller process</a></li>

<li><a href="Fleming%E2%80%93Viot_process" title="Fleming–Viot process">Fleming–Viot process</a></li>

<li><a href="Gamma_process" title="Gamma process">Gamma process</a></li>

<li><a href="Geometric_process" title="Geometric process">Geometric process</a></li>

<li><a href="Hawkes_process" title="Hawkes process">Hawkes process</a></li>

<li><a href="Hunt_process" title="Hunt process">Hunt process</a></li>

<li><a href="Interacting_particle_system" title="Interacting particle system">Interacting particle systems</a></li>

<li><a href="It%C3%B4_diffusion" title="Itô diffusion">Itô diffusion</a></li>

<li><a href="It%C3%B4_process" class="mw-redirect" title="Itô process">Itô process</a></li>

<li><a href="Jump_diffusion" title="Jump diffusion">Jump diffusion</a></li>

<li><a href="Jump_process" title="Jump process">Jump process</a></li>

<li><a href="L%C3%A9vy_process" title="Lévy process">Lévy process</a></li>

<li><a href="Local_time_(mathematics)" title="Local time (mathematics)">Local time</a></li>

<li><a href="Markov_additive_process" title="Markov additive process">Markov additive process</a></li>

<li><a href="McKean%E2%80%93Vlasov_process" title="McKean–Vlasov process">McKean–Vlasov process</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson process</a>

<ul><li><a href="Compound_Poisson_process" title="Compound Poisson process">Compound</a></li>

<li><a href="Non-homogeneous_Poisson_process" class="mw-redirect" title="Non-homogeneous Poisson process">Non-homogeneous</a></li></ul></li>

<li><a href="Quasimartingale" title="Quasimartingale">Quasimartingale</a></li>

<li><a href="Schramm%E2%80%93Loewner_evolution" title="Schramm–Loewner evolution">Schramm–Loewner evolution</a></li>

<li><a href="Semimartingale" title="Semimartingale">Semimartingale</a></li>

<li><a href="Sigma-martingale" title="Sigma-martingale">Sigma-martingale</a></li>

<li><a href="Stable_process" title="Stable process">Stable process</a></li>

<li><a href="Superprocess" title="Superprocess">Superprocess</a></li>

<li><a href="Telegraph_process" title="Telegraph process">Telegraph process</a></li>

<li><a href="Variance_gamma_process" title="Variance gamma process">Variance gamma process</a></li>

<li><a href="Wiener_process" title="Wiener process">Wiener process</a></li>

<li><a href="Wiener_sausage" title="Wiener sausage">Wiener sausage</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Both</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Branching_process" title="Branching process">Branching process</a></li>

<li><a href="Gaussian_process" title="Gaussian process">Gaussian process</a></li>

<li><a href="Hidden_Markov_model" title="Hidden Markov model">Hidden Markov model (HMM)</a></li>

<li><a href="Markov_process" class="mw-redirect" title="Markov process">Markov process</a></li>

<li><a href="Martingale_(probability_theory)" title="Martingale (probability theory)">Martingale</a>

<ul><li><a href="Martingale_difference_sequence" title="Martingale difference sequence">Differences</a></li>

<li><a href="Local_martingale" title="Local martingale">Local</a></li>

<li><a href="Supermartingale" class="mw-redirect" title="Supermartingale">Super-</a></li></ul></li>

<li><a href="Random_dynamical_system" title="Random dynamical system">Random dynamical system</a></li>

<li><a href="Regenerative_process" title="Regenerative process">Regenerative process</a></li>

<li><a href="Renewal_process" class="mw-redirect" title="Renewal process">Renewal process</a></li>

<li><a href="Stochastic_chains_with_memory_of_variable_length" title="Stochastic chains with memory of variable length">Stochastic chains with memory of variable length</a></li>

<li><a href="White_noise" title="White noise">White noise</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Fields and other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li>

<li><a href="Gaussian_random_field" title="Gaussian random field">Gaussian random field</a></li>

<li><a href="Gibbs_measure" title="Gibbs measure">Gibbs measure</a></li>

<li><a href="Hopfield_model" class="mw-redirect" title="Hopfield model">Hopfield model</a></li>

<li><a href="Ising_model" title="Ising model">Ising model</a>

<ul><li><a href="Potts_model" title="Potts model">Potts model</a></li>

<li><a href="Boolean_network" title="Boolean network">Boolean network</a></li></ul></li>

<li><a href="Markov_random_field" title="Markov random field">Markov random field</a></li>

<li><a href="Percolation_theory" title="Percolation theory">Percolation</a></li>

<li><a href="Pitman%E2%80%93Yor_process" title="Pitman–Yor process">Pitman–Yor process</a></li>

<li><a href="Point_process" title="Point process">Point process</a>

<li><a href="Determinantal_point_process" title="Determinantal point process">Determinantal</a></li>

<li><a href="Poisson_point_process" title="Poisson point process">Poisson</a></li></ul></li>

<li><a href="Random_field" title="Random field">Random field</a></li>

<li><a href="Random_graph" title="Random graph">Random graph</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Time_series" title="Time series">Time series models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Autoregressive conditional heteroskedasticity (ARCH) model</a></li>

<li><a href="Autoregressive_integrated_moving_average" title="Autoregressive integrated moving average">Autoregressive integrated moving average (ARIMA) model</a></li>

<li><a href="Autoregressive_model" title="Autoregressive model">Autoregressive (AR) model</a></li>

<li><a href="Autoregressive%E2%80%93moving-average_model" class="mw-redirect" title="Autoregressive–moving-average model">Autoregressive–moving-average (ARMA) model</a></li>

<li><a href="Autoregressive_conditional_heteroskedasticity" title="Autoregressive conditional heteroskedasticity">Generalized autoregressive conditional heteroskedasticity (GARCH) model</a></li>

<li><a href="Moving-average_model" title="Moving-average model">Moving-average (MA) model</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Asset_pricing_model" class="mw-redirect" title="Asset pricing model">Financial models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Binomial_options_pricing_model" title="Binomial options pricing model">Binomial options pricing model</a></li>

<li><a href="Black%E2%80%93Derman%E2%80%93Toy_model" title="Black–Derman–Toy model">Black–Derman–Toy</a></li>

<li><a href="Black%E2%80%93Karasinski_model" title="Black–Karasinski model">Black–Karasinski</a></li>

<li><a href="Black%E2%80%93Scholes_model" title="Black–Scholes model">Black–Scholes</a></li>

<li><a href="Chan%E2%80%93Karolyi%E2%80%93Longstaff%E2%80%93Sanders_process" title="Chan–Karolyi–Longstaff–Sanders process">Chan–Karolyi–Longstaff–Sanders (CKLS)</a></li>

<li><a href="Constant_elasticity_of_variance_model" title="Constant elasticity of variance model">Constant elasticity of variance (CEV)</a></li>

<li><a href="Cox%E2%80%93Ingersoll%E2%80%93Ross_model" title="Cox–Ingersoll–Ross model">Cox–Ingersoll–Ross (CIR)</a></li>

<li><a href="Garman%E2%80%93Kohlhagen_model" class="mw-redirect" title="Garman–Kohlhagen model">Garman–Kohlhagen</a></li>

<li><a href="Heath%E2%80%93Jarrow%E2%80%93Morton_framework" title="Heath–Jarrow–Morton framework">Heath–Jarrow–Morton (HJM)</a></li>

<li><a href="Heston_model" title="Heston model">Heston</a></li>

<li><a href="Hull%E2%80%93White_model" title="Hull–White model">Hull–White</a></li>

<li><a href="Korn%E2%80%93Kreer%E2%80%93Lenssen_model" title="Korn–Kreer–Lenssen model">Korn-Kreer-Lenssen</a></li>

<li><a href="LIBOR_market_model" title="LIBOR market model">LIBOR market</a></li>

<li><a href="Rendleman%E2%80%93Bartter_model" title="Rendleman–Bartter model">Rendleman–Bartter</a></li>

<li><a href="SABR_volatility_model" title="SABR volatility model">SABR volatility</a></li>

<li><a href="Vasicek_model" title="Vasicek model">Vašíček</a></li>

<li><a href="Wilkie_investment_model" title="Wilkie investment model">Wilkie</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial models</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="B%C3%BChlmann_model" title="Bühlmann model">Bühlmann</a></li>

<li><a href="Cram%C3%A9r%E2%80%93Lundberg_model" class="mw-redirect" title="Cramér–Lundberg model">Cramér–Lundberg</a></li>

<li><a href="Risk_process" class="mw-redirect" title="Risk process">Risk process</a></li>

<li><a href="Sparre%E2%80%93Anderson_model" class="mw-redirect" title="Sparre–Anderson model">Sparre–Anderson</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Queueing_model" class="mw-redirect" title="Queueing model">Queueing models</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Fluid_queue" title="Fluid queue">Fluid</a></li>

<li><a href="G-network" title="G-network">Generalized queueing network</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="C%C3%A0dl%C3%A0g" title="Càdlàg">Càdlàg paths</a></li>

<li><a href="Continuous_stochastic_process" title="Continuous stochastic process">Continuous</a></li>

<li><a href="Sample-continuous_process" title="Sample-continuous process">Continuous paths</a></li>

<li><a href="Ergodicity" title="Ergodicity">Ergodic</a></li>

<li><a href="Exchangeable_random_variables" title="Exchangeable random variables">Exchangeable</a></li>

<li><a href="Feller-continuous_process" title="Feller-continuous process">Feller-continuous</a></li>

<li><a href="Gauss%E2%80%93Markov_process" title="Gauss–Markov process">Gauss–Markov</a></li>

<li><a href="Markov_property" title="Markov property">Markov</a></li>

<li><a href="Mixing_(mathematics)" title="Mixing (mathematics)">Mixing</a></li>

<li><a href="Piecewise-deterministic_Markov_process" title="Piecewise-deterministic Markov process">Piecewise-deterministic</a></li>

<li><a href="Predictable_process" title="Predictable process">Predictable</a></li>

<li><a href="Progressively_measurable_process" title="Progressively measurable process">Progressively measurable</a></li>

<li><a href="Self-similar_process" title="Self-similar process">Self-similar</a></li>

<li><a href="Stationary_process" title="Stationary process">Stationary</a></li>

<li><a href="Time_reversibility" title="Time reversibility">Time-reversible</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Limit theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Central_limit_theorem" title="Central limit theorem">Central limit theorem</a></li>

<li><a href="Donsker's_theorem" title="Donsker's theorem">Donsker's theorem</a></li>

<li><a href="Doob's_martingale_convergence_theorems" title="Doob's martingale convergence theorems">Doob's martingale convergence theorems</a></li>

<li><a href="Ergodic_theorem" class="mw-redirect" title="Ergodic theorem">Ergodic theorem</a></li>

<li><a href="Fisher%E2%80%93Tippett%E2%80%93Gnedenko_theorem" title="Fisher–Tippett–Gnedenko theorem">Fisher–Tippett–Gnedenko theorem</a></li>

<li><a href="Large_deviation_principle" class="mw-redirect" title="Large deviation principle">Large deviation principle</a></li>

<li><a href="Law_of_large_numbers" title="Law of large numbers">Law of large numbers (weak/strong)</a></li>

<li><a href="Law_of_the_iterated_logarithm" title="Law of the iterated logarithm">Law of the iterated logarithm</a></li>

<li><a href="Maximal_ergodic_theorem" title="Maximal ergodic theorem">Maximal ergodic theorem</a></li>

<li><a href="Sanov's_theorem" title="Sanov's theorem">Sanov's theorem</a></li>

<li><a href="Zero%E2%80%93one_law" title="Zero–one law">Zero–one laws</a> (<a href="Blumenthal's_zero%E2%80%93one_law" title="Blumenthal's zero–one law">Blumenthal</a>, <a href="Borel%E2%80%93Cantelli_lemma" title="Borel–Cantelli lemma">Borel–Cantelli</a>, <a href="Engelbert%E2%80%93Schmidt_zero%E2%80%93one_law" title="Engelbert–Schmidt zero–one law">Engelbert–Schmidt</a>, <a href="Hewitt%E2%80%93Savage_zero%E2%80%93one_law" title="Hewitt–Savage zero–one law">Hewitt–Savage</a>, <a href="Kolmogorov's_zero%E2%80%93one_law" title="Kolmogorov's zero–one law"> Kolmogorov</a>, <a href="L%C3%A9vy's_zero%E2%80%93one_law" class="mw-redirect" title="Lévy's zero–one law">Lévy</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="List_of_inequalities#Probability_theory_and_statistics" title="List of inequalities">Inequalities</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Burkholder%E2%80%93Davis%E2%80%93Gundy_inequalities" class="mw-redirect" title="Burkholder–Davis–Gundy inequalities">Burkholder–Davis–Gundy</a></li>

<li><a href="Doob's_martingale_inequality" title="Doob's martingale inequality">Doob's martingale</a></li>

<li><a href="Doob's_upcrossing_inequality" class="mw-redirect" title="Doob's upcrossing inequality">Doob's upcrossing</a></li>

<li><a href="Kunita%E2%80%93Watanabe_inequality" title="Kunita–Watanabe inequality">Kunita–Watanabe</a></li>

<li><a href="Marcinkiewicz%E2%80%93Zygmund_inequality" title="Marcinkiewicz–Zygmund inequality">Marcinkiewicz–Zygmund</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cameron%E2%80%93Martin_formula" class="mw-redirect" title="Cameron–Martin formula">Cameron–Martin formula</a></li>

<li><a href="Convergence_of_random_variables" title="Convergence of random variables">Convergence of random variables</a></li>

<li><a href="Dol%C3%A9ans-Dade_exponential" title="Doléans-Dade exponential">Doléans-Dade exponential</a></li>

<li><a href="Doob_decomposition_theorem" title="Doob decomposition theorem">Doob decomposition theorem</a></li>

<li><a href="Doob%E2%80%93Meyer_decomposition_theorem" title="Doob–Meyer decomposition theorem">Doob–Meyer decomposition theorem</a></li>

<li><a href="Doob's_optional_stopping_theorem" class="mw-redirect" title="Doob's optional stopping theorem">Doob's optional stopping theorem</a></li>

<li><a href="Dynkin's_formula" title="Dynkin's formula">Dynkin's formula</a></li>

<li><a href="Feynman%E2%80%93Kac_formula" title="Feynman–Kac formula">Feynman–Kac formula</a></li>

<li><a href="Filtration_(probability_theory)" title="Filtration (probability theory)">Filtration</a></li>

<li><a href="Girsanov_theorem" title="Girsanov theorem">Girsanov theorem</a></li>

<li><a href="Infinitesimal_generator_(stochastic_processes)" title="Infinitesimal generator (stochastic processes)">Infinitesimal generator</a></li>

<li><a href="It%C3%B4_integral" class="mw-redirect" title="Itô integral">Itô integral</a></li>

<li><a href="It%C3%B4's_lemma" title="Itô's lemma">Itô's lemma</a></li>

<li><a href="Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">Karhunen–Loève theorem</a></li>

<li><a href="Kolmogorov_continuity_theorem" title="Kolmogorov continuity theorem">Kolmogorov continuity theorem</a></li>

<li><a href="Kolmogorov_extension_theorem" title="Kolmogorov extension theorem">Kolmogorov extension theorem</a></li>

<li><a href="L%C3%A9vy%E2%80%93Prokhorov_metric" title="Lévy–Prokhorov metric">Lévy–Prokhorov metric</a></li>

<li><a href="Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a></li>

<li><a href="Martingale_representation_theorem" title="Martingale representation theorem">Martingale representation theorem</a></li>

<li><a href="Optional_stopping_theorem" title="Optional stopping theorem">Optional stopping theorem</a></li>

<li><a href="Prokhorov's_theorem" title="Prokhorov's theorem">Prokhorov's theorem</a></li>

<li><a href="Quadratic_variation" title="Quadratic variation">Quadratic variation</a></li>

<li><a href="Reflection_principle_(Wiener_process)" title="Reflection principle (Wiener process)">Reflection principle</a></li>

<li><a href="Skorokhod_integral" title="Skorokhod integral">Skorokhod integral</a></li>

<li><a href="Skorokhod's_representation_theorem" title="Skorokhod's representation theorem">Skorokhod's representation theorem</a></li>

<li><a href="Skorokhod_space" class="mw-redirect" title="Skorokhod space">Skorokhod space</a></li>

<li><a href="Snell_envelope" title="Snell envelope">Snell envelope</a></li>

<li><a href="Stochastic_differential_equation" title="Stochastic differential equation">Stochastic differential equation</a>

<ul><li><a href="Tanaka_equation" title="Tanaka equation">Tanaka</a></li></ul></li>

<li><a href="Stopping_time" title="Stopping time">Stopping time</a></li>

<li><a href="Stratonovich_integral" title="Stratonovich integral">Stratonovich integral</a></li>

<li><a href="Uniform_integrability" title="Uniform integrability">Uniform integrability</a></li>

<li><a href="Usual_hypotheses" class="mw-redirect" title="Usual hypotheses">Usual hypotheses</a></li>

<li><a href="Wiener_space" class="mw-redirect" title="Wiener space">Wiener space</a>

<ul><li><a href="Classical_Wiener_space" title="Classical Wiener space">Classical</a></li>

<li><a href="Abstract_Wiener_space" title="Abstract Wiener space">Abstract</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Disciplines</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Actuarial_mathematics" class="mw-redirect" title="Actuarial mathematics">Actuarial mathematics</a></li>

<li><a href="Stochastic_control" title="Stochastic control">Control theory</a></li>

<li><a href="Econometrics" title="Econometrics">Econometrics</a></li>

<li><a href="Ergodic_theory" title="Ergodic theory">Ergodic theory</a></li>

<li><a href="Extreme_value_theory" title="Extreme value theory">Extreme value theory (EVT)</a></li>

<li><a href="Large_deviations_theory" title="Large deviations theory">Large deviations theory</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li><a href="Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability theory</a></li>

<li><a href="Queueing_theory" title="Queueing theory">Queueing theory</a></li>

<li><a href="Renewal_theory" title="Renewal theory">Renewal theory</a></li>

<li><a href="Ruin_theory" title="Ruin theory">Ruin theory</a></li>

<li><a href="Signal_processing" title="Signal processing">Signal processing</a></li>

<li><a href="Statistics" title="Statistics">Statistics</a></li>

<li><a href="Stochastic_analysis" class="mw-redirect" title="Stochastic analysis">Stochastic analysis</a></li>

<li><a href="Time_series_analysis" class="mw-redirect" title="Time series analysis">Time series analysis</a></li>

<li><a href="Machine_learning" title="Machine learning">Machine learning</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div>

<ul><li><a href="List_of_stochastic_processes_topics" title="List of stochastic processes topics">List of topics</a></li>

<li>Category</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-07" href="https://en.wikipedia.org/wiki/?title=Ornstein%E2%80%93Uhlenbeck_process&oldid=1299267229">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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