analytic solution to Ornstein-Uhlenbeck SDE

This entry derives the analytical solution to the stochastic differential equation for the Ornstein-Uhlenbeck process:

dXt=κ(θ-Xt)dt+σdWt, (1)

where Wt is a standard Brownian motionMathworldPlanetmath, and κ>0, θ, and σ>0 are constants.

Motivated by the observation that θ is supposed to be the long-term mean of the process Xt, we can simplify the SDE (1) by introducing the change of variable


that subtracts off the mean. Then Yt satisfies the SDE:

dYt=dXt=-κYtdt+σdWt. (2)

In SDE (2), the process Yt is seen to have a drift towards the value zero, at an exponentialMathworldPlanetmathPlanetmath rate κ. This motivates the change of variables


which should remove the drift. A calculation with the product ruleMathworldPlanetmath for Itô integralsDlmfPlanetmath shows that this is so:

dZt =κeκtYtdt+eκtdYt

The solution for Zt is immediately obtained by Itô-integrating both sides from s to t:


Reversing the changes of variables, we have:



Title analytic solution to Ornstein-Uhlenbeck SDE
Canonical name AnalyticSolutionToOrnsteinUhlenbeckSDE
Date of creation 2013-03-22 17:19:29
Last modified on 2013-03-22 17:19:29
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 4
Author stevecheng (10074)
Entry type Derivation
Classification msc 60H10
Classification msc 60-00