Definition

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

(1)

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

or

(2)

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.

Bibliography

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.