Ornstein-Uhlenbeck process


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

dXt=κ(θ-Xt)dt+σdWt, (1)

where Wt is a standard Brownian motionMathworldPlanetmath on t[0,).

The constant parameters are:

  • κ>0 is the rate of mean reversion;

  • θ is the long-term mean of the process;

  • σ>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 dWt, then we see that Xt has an overall drift towards a mean value θ. The process Xt reverts to this mean exponentially, at rate κ, with a magnitude in direct proportion to the distance between the current value of Xt and θ.

This can be seen by looking at the solution to the ordinary differential equationMathworldPlanetmath dxt=κ(θ-x)dt which is

θ-xtθ-x0=e-κ(t-t0), or xt=θ+(x0-θ)e-κ(t-t0). (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.


The solution to the stochastic differential equation (1) defining the Ornstein-Uhlenbeck process is, for any 0st, is


where the integral on the right is the Itô integral.

For any fixed s and t, the random variableMathworldPlanetmath Xt, conditionalMathworldPlanetmathPlanetmath upon Xs, is normally distributed with


Observe that the mean of Xt 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.


  • 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
Canonical name OrnsteinUhlenbeckProcess
Date of creation 2013-03-22 17:19:26
Last modified on 2013-03-22 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 60-00
Synonym Ornstein-Uhlenbeck equation