奧恩斯坦-烏倫貝克過程

在數學中，奧恩斯坦-烏倫貝克過程（Ornstein-Uhlenbeck process，簡稱OU過程）是一個隨機過程，在金融數學和物理學中有很多的引用。OU過程描述一個經歷摩擦的布朗粒子（damped random walk）。^[1]

這個過程以奧恩斯坦（Leonard Ornstein）和喬治·烏倫貝克的名字命名。

這是一個自迴歸模型AR(1)。

θ =1.0，σ =3和μ =(0,0) 粒子在(10,10)開始

定義

θ =1.0，σ =3, μ =(0,0,0) 粒子在(10,10,10)開始

OU過程有下面的隨機微分方程

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

其中的 ${\displaystyle \theta >0}$ ， ${\displaystyle \sigma >0}$ 是參數，並且 ${\displaystyle W_{t}}$ 是維納過程。^[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]

福克–普朗克方程

OU過程的福克–普朗克方程是^[6]

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

${\displaystyle D=\sigma ^{2}/2}$。這是一個拋物偏微分方程。方程的解是

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

三個OU進程，θ = 1, μ = 1.2, σ = 0.3:
藍：在a = 0 開始（幾乎必然）
綠：在a=2開始
紅：初始值呈常態分布

相關

• CKLS過程^[7]（Chan–Karolyi–Longstaff–Sanders process）
• 陳模型
• 縮放極限

參考文獻

1. MacLeod, C. L.; Ivezić, Ž; Kochanek, C. S.; Kozłowski, S.; Kelly, B.; Bullock, E.; Kimball, A.; Sesar, B.; Westman, D. Modeling the Time Variability of SDSS Stripe 82 Quasars as a Damped Random Walk. The Astrophysical Journal. October 2010, 721: 1014. doi:10.1088/0004-637X/721/2/1014 （英語）.^{[永久失效連結]}
2. Karatzas, Ioannis; Shreve, Steven E., Brownian Motion and Stochastic Calculus 2nd, Springer-Verlag: 358, 1991, ISBN 978-0-387-97655-6
3. Gard, Thomas C., Introduction to Stochastic Differential Equations, Marcel Dekker: 115, 1988, ISBN 978-0-8247-7776-0
4. Gardiner, C.W., Handbook of Stochastic Methods 2nd, Springer-Verlag: 106, 1985, ISBN 978-0-387-15607-1
5. Björk, Tomas. Arbitrage Theory in Continuous Time 3rd. Oxford University Press. 2009: 375, 381. ISBN 978-0-19-957474-2.
6. Risken, H., The Fokker-Planck Equation: Methods of Solution and Application, Springer-Verlag: 99–100, 1984, ISBN 978-0-387-13098-9
7. Chan et al. (1992)

閱讀

• Bibbona, E.; Panfilo, G.; Tavella, P. The Ornstein-Uhlenbeck process as a model of a low pass filtered white noise. Metrologia. 2008, 45 (6): S117–S126. Bibcode:2008Metro..45S.117B. doi:10.1088/0026-1394/45/6/S17.
• Chan, K. C.; Karolyi, G. A.; Longstaff, F. A.; Sanders, A. B. An empirical comparison of alternative models of the short-term interest rate. Journal of Finance. 1992, 47 (3): 1209–1227. doi:10.1111/j.1540-6261.1992.tb04011.x.
• Doob, J.L. The Brownian Movement and Stochastic Equations. Annals of Mathematics. April 1942, 43 (2): 351–369. JSTOR 1968873. doi:10.2307/1968873.
• Gillespie, D. T. Exact numerical simulation of the Ornstein–Uhlenbeck process and its integral (PDF). Phys. Rev. E. 1996, 54 (2): 2084–2091 [2020-02-11]. Bibcode:1996PhRvE..54.2084G. PMID 9965289. doi:10.1103/PhysRevE.54.2084. （原始內容 (PDF)存檔於2022-03-01）.
• Leung, Tim; Li, Xin. Optimal Mean Reversion Trading with Transaction Costs and Stop-Loss Exit. International Journal of Theoretical & Applied Finance. 2015, 18 (3): 1550020. arXiv:1411.5062 . doi:10.1142/S021902491550020X.
• Risken, H. The Fokker–Planck Equation: Method of Solution and Applications. New York: Springer-Verlag. 1989. ISBN 978-0387504988.
• Uhlenbeck, G. E.; Ornstein, L. S. On the theory of Brownian Motion. Phys. Rev. 1930, 36 (5): 823–841. Bibcode:1930PhRv...36..823U. doi:10.1103/PhysRev.36.823.
• Martins, E.P. Estimating the Rate of Phenotypic Evolution from Comparative Data. Amer. Nat. 1994, 144 (2): 193–209.

外部連結

• Review of Statistical Arbitrage, Cointegration, and Multivariate Ornstein–Uhlenbeck （頁面存檔備份，存於網際網路檔案館）, 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 （頁面存檔備份，存於網際網路檔案館）, Jose Carlos Garcia Franco
• Interactive Web Application: Stochastic Processes used in Quantitative Finance. [2015-07-03]. （原始內容存檔於2015-09-20）.