Ornstein-Uhlenbeck process

Definition

The Ornstein-Uhlenbeck process is a stochastic processthat satisfies the following stochastic differential equation:

$d{X}_t=\kappa (\theta -X_t)dt+\sigma d{W}_t,$(1)

where $W_t$ is a standard Brownian motion on $t\in [0,\mathrm{\infty })$.

The constant parameters are:

• $\kappa >0$ is the rate of mean reversion;

• $\theta$ is the long-term mean of the process;

• $\sigma >0$ is the volatility or average magnitude, per square-root time,of the random fluctuations that are modelled as Brownian motions.

Mean-reverting property

If we ignore the random fluctuations in the processdue to $d{W}_t$, then we see that $X_t$ has an overall drifttowards a mean value $\theta$.The process $X_t$ reverts to this mean exponentially, at rate $\kappa$,with a magnitude in direct proportion to the distancebetween the current value of $X_t$ and $\theta$.

This can be seen by looking at the solution to theordinary differential equation $d{x}_t=\kappa (\theta -x)dt$which is

${\displaystyle \frac{\theta -x_t}{\theta -x_0}}=e^{-\kappa (t-t_0)},\text{ or }{x}_t=\theta +(x_0-\theta )e^{-\kappa (t-t_0)}.$(2)

For this reason, the Ornstein-Uhlenbeck processis also called a mean-reverting process,although the latter name applies to other typesof stochastic processes exhibiting the same property as well.

Solution

The solution to the stochastic differential equation (1)defining the Ornstein-Uhlenbeck process is, for any $0\le s\le t$, is

$$X_t=\theta +(X_s-\theta )e^{-\kappa (t-s)}+\sigma {\int }_s^t{e}^{-\kappa (t-u)}𝑑W_u.$$

where the integral on the right is the Itô integral.

For any fixed $s$ and $t$, the random variable $X_t$, conditionalupon $X_s$, is normally distributed with

$$\text{mean}=\theta +(X_s-\theta )e^{-\kappa (t-s)},\text{variance}=\frac{{\sigma }^2}{2\kappa }(1-e^{-2\kappa (t-s)}).$$

Observe that the mean of $X_t$ is exactlythe value derived heuristicallyin the solution (2) of the ODE.

The Ornstein-Uhlenbeck process is a time-homogeneous Itô diffusion.

Applications

The Ornstein-Uhlenbeck process is widely used for modellingbiological processes such as neuronal response,and in mathematical finance,the modelling of the dynamics of interest ratesand volatilities of asset prices.