An analytical characterization for an optimal change of Gaussian measures

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.