奥恩斯坦-乌伦贝克过程 - 维基百科，自由的百科全书

在数学中，奥恩斯坦-乌伦贝克过程（Ornstein-Uhlenbeck process，简称OU过程）是一个随机过程，在金融数学和物理学中有很多的引用。OU过程描述一个经历摩擦的布朗粒子（damped random walk）。^[1]

这个过程以奥恩斯坦（Leonard Ornstein）和乔治·乌伦贝克的名字命名。

这是一个自迴歸模型AR(1)。

OU过程有下面的随机微分方程

${\displaystyle dx_{t}=-\theta \,x_{t}\,dt+\sigma \,dW_{t}}$

其中的 ${\displaystyle \theta >0}$ ， ${\displaystyle \sigma >0}$ 是参数，并且 ${\displaystyle W_{t}}$ 是维纳过程。^[2][3][4]

${\displaystyle dx_{t}=\theta (\mu -x_{t})\,dt+\sigma \,dW_{t}}$

${\displaystyle \mu }$ 是常值。上面的方程是Vasicek模型。^[5]

OU过程的福克–普朗克方程是^[6]

${\displaystyle {\frac {\partial P}{\partial t}}=\theta {\frac {\partial }{\partial x}}(xP)+D{\frac {\partial ^{2}P}{\partial x^{2}}}}$

${\displaystyle D=\sigma ^{2}/2}$。这是一个抛物偏微分方程。方程的解是

${\displaystyle P(x,t\mid x',t')={\sqrt {\frac {\theta }{2\pi D(1-e^{-2\theta (t-t')})}}}\exp \left[-{\frac {\theta }{2D}}{\frac {(x-x'e^{-\theta (t-t')})^{2}}{1-e^{-2\theta (t-t')}}}\right]}$

• CKLS过程^[7]（Chan–Karolyi–Longstaff–Sanders process）
• 陈模型
• 缩放极限

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• Review of Statistical Arbitrage, Cointegration, and Multivariate Ornstein–Uhlenbeck （页面存档备份，存于互联网档案馆）, Attilio Meucci
• A Stochastic Processes Toolkit for Risk Management, Damiano Brigo, Antonio Dalessandro, Matthias Neugebauer and Fares Triki
• Simulating and Calibrating the Ornstein–Uhlenbeck process, M. A. van den Berg
• Maximum likelihood estimation of mean reverting processes （页面存档备份，存于互联网档案馆）, Jose Carlos Garcia Franco
• Interactive Web Application: Stochastic Processes used in Quantitative Finance. [2015-07-03]. （原始内容存档于2015-09-20）.