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
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
其中的
,
是參數,並且
是維納過程。[2][3][4]
是常值。上面的方程是Vasicek模型。[5]
OU過程的福克–普朗克方程是[6]
。這是一個拋物偏微分方程。方程的解是
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
- CKLS過程[7](Chan–Karolyi–Longstaff–Sanders process)
- 陳模型
- 縮放極限
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
Risken, H., The Fokker-Planck Equation: Methods of Solution and Application, Springer-Verlag: 99–100, 1984, ISBN 978-0-387-13098-9
- 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.