# Ornstein-Uhlenbeck process

## Definition

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

 $\displaystyle dX_{t}=\kappa(\theta-X_{t})\,dt+\sigma\,dW_{t}\,,$ (1)

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.

## Mean-reverting property

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

 $\displaystyle\frac{\theta-x_{t}}{\theta-x_{0}}=e^{-\kappa(t-t_{0})}\,,\quad% \text{ or }x_{t}=\theta+(x_{0}-\theta)e^{-\kappa(t-t_{0})}\,.$ (2)

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.

## Solution

The solution to the stochastic differential equation (1) defining the Ornstein-Uhlenbeck process is, for any $0\leq s\leq t$, 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 $X_{t}$, conditional upon $X_{s}$, is normally distributed with

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

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.

## Applications

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.

## References

• 1 Martin Jacobsen. “Laplace and the Origin of the Ornstein-Uhlenbeck Process”. Bernoulli, Vol. 2, No. 3. (Sept. 1996), pp. 271 – 286.
• 2 Bernt Øksendal. Stochastic Differential Equations, An Introduction with Applications, 5th edition. Springer, 1998.
• 3 Steven E. Shreve. Stochastic Calculus for Finance II: Continuous-Time Models. Springer, 2004.
• 4 Sebastian Jaimungal. Lecture notes for Pricing Theory. University of Toronto.
• 5 Dmitri Rubisov. Lecture notes for Risk Management. University of Toronto.
Title Ornstein-Uhlenbeck process OrnsteinUhlenbeckProcess 2013-03-22 17:19:26 2013-03-22 17:19:26 stevecheng (10074) stevecheng (10074) 4 stevecheng (10074) Definition msc 60H10 msc 60-00 Ornstein-Uhlenbeck equation