Ornstein-Uhlenbeck Process

Ornstein-Uhlenbeck (OU) Process Definition

The Ornstein-Uhlenbeck process \(\{X_{t}\}\) has dynamics described by the stochastic differential equation

\[ dX_{t}=-\theta X_{t}dt+\sigma dW_{t}, \]

where \(\theta>0\) and \(\sigma>0\) and \(W_{t}\) is a BM.

An additional drift term is sometimes included giving the model the form

\[ dX_{t}=\theta(\mu-X_{t})dt+\sigma dW_{t} \]

where \(\mu\) is a constant.

OU Process Solution

The process can be formally solved by variation of parameters.
We can define \(f(t,X_{t})=e^{\theta t}X_{t}\) and applying [[Itô Formula]] we have that

\[ \begin{align} d(e^{\theta t}X_{t}) & =\theta e^{\theta t}X_{t}dt+e^{\theta t}dX_{t} \\ & =e^{\theta t}\theta \mu dt+\sigma e^{\theta t}dW_{t} . \end{align} \]

Integrating from \(0\) to \(t\) we get

\[ \begin{align} X_{t}e^{\theta t} & = X_{0}+\int _{0}^te^{\theta s}\theta \mu \, ds + \int _{0}^t\sigma e^{\theta s} \, dW_{s} \\ \implies X_{t} & = X_{0}e^{-\theta t}+\mu(1-e^{-\theta t})+ \sigma \int _{0}^te^{-\theta(t-s)} \, dW_{s.} \end{align} \]

Langevin Equation Form

The process is sometimes also written as a [[Langevin Equation]] of the form

\[ \frac{dX_{t}}{dt}=-\theta X_{t}+\sigma \eta(t), \]

where \(\eta(t)\), also known as white noise, stands in for the supposed derivative \(\frac{dW_{t}}{dt}\) of BM - which doesn’t exist as BM is nowhere differentiable, this is a common representation by tacitly assuming that the noise term is a derivative of a differentiable (e.g. Fourier) interpolation of BM.

Kolmogorov Equation

The infinitesimal generator of the process is

\[ Lf=-\theta(x-\mu)f'+ \frac{1}{2}\sigma^2 f''. \]