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

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

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.

If we ignore the random fluctuations in the process due to $dW_{t}$, then we see that $X_{t}$ has an overall drift towards a mean value $\theta$. The process $X_{t}$ reverts to this mean exponentially, at rate $\kappa$, with a magnitude in direct proportion to the distance between the current value of $X_{t}$ and $\theta$.

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

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

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

where the integral on the right is the Itô integral.

For any fixed $s$ and $t$, the random variable $X_{t}$, conditional upon $X_{s}$, is normally distributed with

Observe that the mean of $X_{t}$ is exactly the value derived heuristically in the solution (2) of the ODE.

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

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