查看“︁奥恩斯坦-乌伦贝克过程”︁的源代码

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[[布朗运动|布朗粒子]]（damped random walk）。<ref>{{Cite journal|title=Modeling the Time Variability of SDSS Stripe 82 Quasars as a Damped Random Walk|url=https://ui.adsabs.harvard.edu/2010ApJ...721.1014M/abstract|last=MacLeod|first=C. L.|last2=Ivezić|first2=Ž|date=October 2010|journal=The Astrophysical Journal|doi=10.1088/0004-637X/721/2/1014|volume=721|pages=1014|language=en|last3=Kochanek|first3=C. S.|last4=Kozłowski|first4=S.|last5=Kelly|first5=B.|last6=Bullock|first6=E.|last7=Kimball|first7=A.|last8=Sesar|first8=B.|last9=Westman|first9=D.}}{{Dead link|date=2020年4月 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

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)。
[[File:OrnsteinUhlenbeckProcess2D.svg|链接=https://zh.wikipedia.org/wiki/File:OrnsteinUhlenbeckProcess2D.svg|缩略图|''θ'' =1.0，''σ'' =3和'''μ''' =(0,0)

粒子在(10,10)开始]]

== 定义 ==
[[File:OrnsteinUhlenbeckProcess3D.svg|链接=https://zh.wikipedia.org/wiki/File:OrnsteinUhlenbeckProcess3D.svg|缩略图|''θ'' =1.0，''σ'' =3, '''μ''' =(0,0,0)

粒子在(10,10,10)开始]]
OU过程有下面的[[隨機微分方程|随机微分方程]]

<math>dx_t = -\theta \, x_t \, dt + \sigma \, dW_t</math>

其中的 <math>\theta > 0</math> ， <math>\sigma > 0</math> 是参数，并且 <math>W_t</math> 是[[维纳过程]]。<ref>{{Citation|last=Karatzas|first=Ioannis|last2=Shreve|first2=Steven E.|title=Brownian Motion and Stochastic Calculus|publisher=Springer-Verlag|year=1991|isbn=978-0-387-97655-6|page=358|edition=2nd}}</ref><ref>{{Citation|last=Gard|first=Thomas C.|title=Introduction to Stochastic Differential Equations|publisher=Marcel Dekker|year=1988|isbn=978-0-8247-7776-0|page=115}}</ref><ref>{{Citation|last=Gardiner|first=C.W.|title=Handbook of Stochastic Methods|publisher=Springer-Verlag|year=1985|isbn=978-0-387-15607-1|page=106|edition=2nd}}</ref>

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

<math>\mu</math> 是常值。上面的方程是Vasicek模型。<ref>{{Cite book|last=Björk|first=Tomas|title=Arbitrage Theory in Continuous Time|url=https://archive.org/details/arbitragetheoryc00bjor|publisher=Oxford University Press|isbn=978-0-19-957474-2|pages=[https://archive.org/details/arbitragetheoryc00bjor/page/375 375], 381|edition=3rd|year=2009}}</ref>

== 福克–普朗克方程 ==
OU过程的[[福克-普朗克方程|福克–普朗克方程]]是<ref>{{Citation|last=Risken|first=H.|title=The Fokker-Planck Equation: Methods of Solution and Application|publisher=Springer-Verlag|year=1984|isbn=978-0-387-13098-9|pages=99–100}}</ref>

<math>\frac{\partial P}{\partial t} = \theta \frac{\partial}{\partial x} (x P) + D \frac{\partial^2 P}{\partial x^2}</math>

<math>D = \sigma^2 / 2</math>。这是一个[[抛物偏微分方程]]。方程的解是

<math>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]</math>
[[File:OrnsteinUhlenbeck3.png|链接=https://zh.wikipedia.org/wiki/File:OrnsteinUhlenbeck3.png|缩略图|450x450像素
|三个OU进程，''θ''&nbsp;=&nbsp;1, ''μ''&nbsp;=&nbsp;1.2, ''σ''&nbsp;=&nbsp;0.3: 
<br>
<span style="color:#000080;">'''蓝'''</span>：在a&nbsp;=&nbsp;0 开始（[[几乎必然]]）
<br>
<span style="color:#6e8b3d;">'''绿'''</span>：在a=2开始
<br>
<span style="color:#ff0000;">'''红'''</span>：初始值呈正态分布]]

== 相关 ==

* [[CKLS过程]]<ref>Chan et al. (1992)</ref>（Chan–Karolyi–Longstaff–Sanders process）
*[[陈模型]]
*[[缩放极限]]

== 参考文献 ==
<references group="" responsive="1"></references>

== 阅读 ==

* {{Cite journal|title=The Ornstein-Uhlenbeck process as a model of a low pass filtered white noise|last=Bibbona|first=E.|last2=Panfilo|first2=G.|journal=Metrologia|issue=6|doi=10.1088/0026-1394/45/6/S17|year=2008|volume=45|pages=S117–S126|bibcode=2008Metro..45S.117B|last3=Tavella|first3=P.}}
* {{Cite journal|title=An empirical comparison of alternative models of the short-term interest rate|url=https://archive.org/details/sim_journal-of-finance_1992-07_47_3/page/1209|last=Chan|first=K. C.|last2=Karolyi|first2=G. A.|journal=[[Journal of Finance]]|issue=3|doi=10.1111/j.1540-6261.1992.tb04011.x|year=1992|volume=47|pages=1209–1227|last3=Longstaff|first3=F. A.|last4=Sanders|first4=A. B.}}
* {{Cite journal|title=The Brownian Movement and Stochastic Equations|url=https://archive.org/details/sim_annals-of-mathematics_1942-04_43_2/page/351|last=Doob|first=J.L.|authorlink=Joseph Leo Doob|date=April 1942|journal=[[Annals of Mathematics]]|issue=2|doi=10.2307/1968873|volume=43|pages=351–369|jstor=1968873}}
* {{Cite journal|title=Exact numerical simulation of the Ornstein–Uhlenbeck process and its integral|url=https://zenodo.org/record/1233795/files/article.pdf|last=Gillespie|first=D. T.|journal=[[Physical Review|Phys. Rev. E]]|issue=2|doi=10.1103/PhysRevE.54.2084|year=1996|volume=54|pages=2084–2091|bibcode=1996PhRvE..54.2084G|pmid=9965289|access-date=2020-02-11|archive-date=2022-03-01|archive-url=https://web.archive.org/web/20220301215809/https://zenodo.org/record/1233795/files/article.pdf}}
* {{Cite journal|title=Optimal Mean Reversion Trading with Transaction Costs and Stop-Loss Exit|last=Leung|first=Tim|last2=Li|first2=Xin|journal=International Journal of Theoretical & Applied Finance|issue=3|doi=10.1142/S021902491550020X|year=2015|volume=18|pages=1550020|arxiv=1411.5062}}
* {{Cite book|first=H.|last=Risken|title=The Fokker–Planck Equation: Method of Solution and Applications|publisher=Springer-Verlag|location=New York|year=1989|isbn=978-0387504988}}
* {{Cite journal|title=On the theory of Brownian Motion|last=Uhlenbeck|first=G. E.|last2=Ornstein|first2=L. S.|journal=Phys. Rev.|issue=5|doi=10.1103/PhysRev.36.823|year=1930|volume=36|pages=823–841|bibcode=1930PhRv...36..823U}}
* {{Cite journal|title=Estimating the Rate of Phenotypic Evolution from Comparative Data|last=Martins|first=E.P.|journal=Amer. Nat.|issue=2|year=1994|volume=144|pages=193-209}}

==外部链接==
*[http://www.symmys.com/node/132 Review of Statistical Arbitrage, Cointegration, and Multivariate Ornstein–Uhlenbeck] {{Wayback|url=http://www.symmys.com/node/132 |date=20161028151626 }}, Attilio Meucci
*[https://ssrn.com/abstract=1109160 A Stochastic Processes Toolkit for Risk Management], Damiano Brigo, Antonio Dalessandro, Matthias Neugebauer and Fares Triki
*[https://web.archive.org/web/20150619164944/http://www.sitmo.com/article/calibrating-the-ornstein-uhlenbeck-model/ Simulating and Calibrating the Ornstein–Uhlenbeck process], M. A. van den Berg
*[http://www.investmentscience.com/Content/howtoArticles/MLE_for_OR_mean_reverting.pdf Maximum likelihood estimation of mean reverting processes] {{Wayback|url=http://www.investmentscience.com/Content/howtoArticles/MLE_for_OR_mean_reverting.pdf |date=20201203034815 }}, Jose Carlos Garcia Franco
* {{cite web | url=http://turingfinance.com/interactive-stochastic-processes/ | title=Interactive Web Application: Stochastic Processes used in Quantitative Finance | access-date=2015-07-03 | archive-url=https://web.archive.org/web/20150920231636/http://turingfinance.com/interactive-stochastic-processes/ | archive-date=2015-09-20 | url-status=yes }}

{{Stochastic processes}}
[[Category:隨機微分方程]]