Ornstein–Uhlenbeck Process

The Ornstein–Uhlenbeck process combines random fluctuations with a force pulling the particle toward an equilibrium. The model originates in the work of Uhlenbeck and Ornstein [UhlenbeckOrnstein1930].

ORNSTEIN–UHLENBECK PROCESS

Let

W_t

be a standard Brownian motion, with

\theta>0

\sigma>0

, and a fixed initial value

x_0

. The Ornstein–Uhlenbeck process with equilibrium zero solves

\begin{equation*}\begin{cases}\mathrm{d}X_t=-\theta X_t\,\mathrm{d}t+\sigma\,\mathrm{d}W_t,\\X_0=x_0.\end{cases}\end{equation*}

Explicit solution via Itô's formula

We can find the explicit solution by applying Itô’s formula to

e^{\theta t}X_t

Define

\begin{equation*}g(t,x)=e^{\theta t}x.\end{equation*}

Its partial derivatives are

\begin{equation*}g_t(t,x)=\theta e^{\theta t}x,\qquad g_x(t,x)=e^{\theta t},\qquad g_{xx}(t,x)=0.\end{equation*}

Itô's formula gives

\begin{equation*}\mathrm{d}g(t,X_t)=g_t(t,X_t)\,\mathrm{d}t+g_x(t,X_t)\,\mathrm{d}X_t+\underbrace{\frac12\sigma^2g_{xx}(t,X_t)\,\mathrm{d}t}_{\text{Itô's term}}.\end{equation*}

Substituting the OU equation, we obtain

\begin{align*}\mathrm{d}(e^{\theta t}X_t) &= \theta e^{\theta t}X_t\,\mathrm{d}t+e^{\theta t}(-\theta X_t\,\mathrm{d}t+\sigma\,\mathrm{d}W_t)\\&= \sigma e^{\theta t}\,\mathrm{d}W_t.\end{align*}

The drift terms cancel. Integrating from

0

t

and using

X_0=x_0

yields

\begin{equation*}e^{\theta t}X_t-x_0=\sigma\int_0^t e^{\theta s}\,\mathrm{d}W_s.\end{equation*}

Multiplying by

e^{-\theta t}

, we find the explicit solution:

\begin{equation*}\boxed{X_t=x_0e^{-\theta t}+\sigma\int_0^t e^{-\theta(t-s)}\,\mathrm{d}W_s.}\end{equation*}

References

[UhlenbeckOrnstein1930]Uhlenbeck, G. E., and L. S. Ornstein. ``On the Theory of the Brownian Motion.'' Physical Review 36 (1930): 823–841.
[LeGall2016]Le Gall, Jean-François. Brownian Motion, Martingales, and Stochastic Calculus. Springer, 2016.