Journal of Applied Mathematics and Decision Sciences
Volume 2006 (2006), Article ID 95912, 9 pages

An analytical characterization for an optimal change of Gaussian measures

Henry Schellhorn

School of Mathematical Sciences, Claremont Graduate University, Claremont 91711, CA, USA

Received 25 February 2006; Revised 9 June 2006; Accepted 9 June 2006

Copyright © 2006 Henry Schellhorn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider two Gaussian measures. In the “initial” measure the state variable is Gaussian, with zero drift and time-varying volatility. In the “target measure” the state variable follows an Ornstein-Uhlenbeck process, with a free set of parameters, namely, the time-varying speed of mean reversion. We look for the speed of mean reversion that minimizes the variance of the Radon-Nikodym derivative of the target measure with respect to the initial measure under a constraint on the time integral of the variance of the state variable in the target measure. We show that the optimal speed of mean reversion follows a Riccati equation. This equation can be solved analytically when the volatility curve takes specific shapes. We discuss an application of this result to simulation, which we presented in an earlier article.