Journal of Applied Mathematics and Decision Sciences
Volume 2006 (2006), Article ID 95912, 9 pages
doi:10.1155/JAMDS/2006/95912
An analytical characterization for an optimal change of Gaussian measures
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.
Abstract
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.