# 奥恩斯坦-乌伦贝克过程

（重定向自Ornstein–Uhlenbeck过程

θ =1.0，σ =3和μ =(0,0) 粒子在(10,10)开始

## 定义

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

OU过程有下面的随机微分方程

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

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

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

## 福克–普朗克方程

OU过程的福克–普朗克方程[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

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

：在a = 0 开始（几乎必然

：初始值呈正态分布

## 参考文献

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 （英语）.[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)