Ornstein-Uhlenbeck Process
1 Definition
The Ornstein-Uhlenbeck (OU) process has dynamics described by the stochastic differential equation \[ dX_{t}=-\theta X_{t}dt+\sigma dW_{t}, \] where \(\theta>0\), \(\sigma>0\) and \(W_{t}\) is a BM. It is a mean reverting stochastic process that is a Gaussian process, is Markov and is temporally stationary. This process can be used to describe the fluctuation of interest rate around the mean interest rate \(M\) with the speed of the regression described by \(\theta\).
The Ornstein-Uhlenbeck process \((X_{t})\) with drift has dynamics described by the stochastic differential equation \[ dX_{t}=\theta(\mu-X_{t})dt+\sigma dB_{t} \] where \(\theta,\sigma>0\), \(\mu \in \mathbb{R}\) are all constant and \(B_{t}\) is a BM.
2 Solution
2.1 Change of Variable
Consider such a process with a deterministic starting value \(X_{0}\). To find a solution to this SDE we first change the variable to set the regression level to 0 with \(Y_{t}=X_{t}-\mu\). From Itô’s formula we have \[ dY_{t}=dX_{t}=\theta(\mu-X_{t})dt+\sigma dB_{t}=-\theta Y_{t}dt+\sigma dB_{t}. \] Considering when \(\sigma=0\) we have the ODE \(dY_{t}=-\theta Y_{t}dt\) with the solution \(Y_{t}=Y_{0}\cdot e^{-\theta t}\) and so replacing this constant \(Y_{0}\) with some yet to define process \(C_{t}\) we have \(Y_{t}=C_{t}e^{-\theta t}\) and from Itô’s formula we have \[ dY_{t}=-\theta e^{-\theta t}C_{t}dt+e^{-\theta t}dC_{t}=-\theta Y_{t}dt+e^{-\theta t}dC_{t}. \] We therefore can define \(dC_{t}=\sigma e^{\theta t}dB_{t}\) and so \(C_{0}=Y_{0}=X_{0}-\mu\) and \[ \begin{align} C_{t}-C_{0} & = \int_{0}^t\sigma e^{\theta s}dB_{s} \\ \implies C_{t} & =X_{0}-\mu+\sigma \int_{0}^te^{\theta s}dB_{s} \\ \implies Y_{t} & = (X_{0}-\mu)e^{-\theta t}+\sigma e^{-\theta t}\int_{0}^te^{\theta s}dB_{s} \\ \implies X_{t} & = (X_{0}-\mu)e^{-\theta t}+\mu+\sigma e^{-\theta t}\int_{0}^te^{\theta s}dB_{s}. \end{align} \]
2.2 Variation of Parameters
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
\[ \begin{align} X_{t}e^{\theta t} & = X_{0}+\int _{0}^te^{\theta s}\theta \mu \, ds + \int _{0}^t\sigma e^{\theta s} \, dW_{s} \\ \implies X_{t} & = X_{0}e^{-\theta t}+\mu(1-e^{-\theta t})+ \sigma \int _{0}^te^{-\theta(t-s)} \, dW_{s.} \end{align} \]
3 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.
4 Kolmogorov Equation
The Infinitesimal Generator of the process is
\[ Lf=-\theta(x-\mu)f'+ \frac{1}{2}\sigma^2 f''. \]