where is a standard Brownian motion on .
The constant parameters are:
is the rate of mean reversion;
is the long-term mean of the process;
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 , then we see that has an overall drift towards a mean value . The process reverts to this mean exponentially, at rate , with a magnitude in direct proportion to the distance between the current value of and .
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 , is
where the integral on the right is the Itô integral.
Observe that the mean of 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.
|Date of creation||2013-03-22 17:19:26|
|Last modified on||2013-03-22 17:19:26|
|Last modified by||stevecheng (10074)|