OrnsteinUhlenbeck process
Definition
The OrnsteinUhlenbeck process is a stochastic process^{} that 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 longterm mean of the process;

•
$\sigma >0$ is the volatility or average magnitude, per squareroot time, of the random fluctuations that are modelled as Brownian motions.
Meanreverting property
If we ignore the random fluctuations in the process due to $d{W}_{t}$, 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^{} $d{x}_{t}=\kappa (\theta x)dt$ which is
$\frac{\theta {x}_{t}}{\theta {x}_{0}}}={e}^{\kappa (t{t}_{0})},\text{or}{x}_{t}=\theta +({x}_{0}\theta ){e}^{\kappa (t{t}_{0})}.$  (2) 
For this reason, the OrnsteinUhlenbeck process is also called a meanreverting 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 OrnsteinUhlenbeck process is, for any $0\le s\le t$, is
$${X}_{t}=\theta +({X}_{s}\theta ){e}^{\kappa (ts)}+\sigma {\int}_{s}^{t}{e}^{\kappa (tu)}\mathit{d}{W}_{u}.$$ 
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
$$\text{mean}=\theta +({X}_{s}\theta ){e}^{\kappa (ts)},\text{variance}=\frac{{\sigma}^{2}}{2\kappa}(1{e}^{2\kappa (ts)}).$$ 
Observe that the mean of ${X}_{t}$ is exactly the value derived heuristically in the solution (2) of the ODE.
The OrnsteinUhlenbeck process is a timehomogeneous Itô diffusion.
Applications
The OrnsteinUhlenbeck 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 OrnsteinUhlenbeck 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: ContinuousTime 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  OrnsteinUhlenbeck process 

Canonical name  OrnsteinUhlenbeckProcess 
Date of creation  20130322 17:19:26 
Last modified on  20130322 17:19:26 
Owner  stevecheng (10074) 
Last modified by  stevecheng (10074) 
Numerical id  4 
Author  stevecheng (10074) 
Entry type  Definition 
Classification  msc 60H10 
Classification  msc 6000 
Synonym  OrnsteinUhlenbeck equation 