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

Yt=Xt-θ

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

Yt=e-κtZtZt=eκtYt,

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

dZt =κeκtYtdt+eκtdYt
=κeκtYtdt+eκt(-κYtdt+σdWt)
=0dt+σeκtdWt.

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

Zt=Zs+σsteκu𝑑Wu.

Reversing the changes of variables, we have:

Yt=e-κtZt=e-κ(t-s)Ys+σe-κtsteκu𝑑Wu,

and

Xt=Yt+θ=θ+e-κ(t-s)(Xs-θ)+σste-κ(t-u)𝑑Wu.
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