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)。

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

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

其中的 $\theta >0$ ， $\sigma >0$ 是參數，並且 $W_{t}$ 是維納過程。^[2]^[3]^[4]

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

$\mu$ 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]

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

$D=\sigma ^{2}/2$ 。這是一個拋物偏微分方程。方程的解是

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

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：初始值呈正態分布

參考文獻

^ 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]}
^ Karatzas, Ioannis; Shreve, Steven E., Brownian Motion and Stochastic Calculus 2nd, Springer-Verlag: 358, 1991, ISBN 978-0-387-97655-6
^ Gard, Thomas C., Introduction to Stochastic Differential Equations, Marcel Dekker: 115, 1988, ISBN 978-0-8247-7776-0
^ Gardiner, C.W., Handbook of Stochastic Methods 2nd, Springer-Verlag: 106, 1985, ISBN 978-0-387-15607-1
^ Björk, Tomas. Arbitrage Theory in Continuous Time 3rd. Oxford University Press. 2009: 375, 381. ISBN 978-0-19-957474-2.
^ Risken, H., The Fokker-Planck Equation: Methods of Solution and Application, Springer-Verlag: 99–100, 1984, ISBN 978-0-387-13098-9
^ 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）.

[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)

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奧恩斯坦-烏倫貝克過程

目次

定義

福克–普朗克方程

相關

參考文獻

閱讀

外部連結