An analytical characterization for an optimal change of Gaussian measures

Henry Schellhorn

doi:10.1155/jamds/2006/95912

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.

Highlights

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 studied this problem in an earlier article, where we explained one application of this result to the field of Monte Carlo simulation
Care must be taken in selecting a target measure for simulation performance, and we suggested that a good performance measure was the variance of g

Summary

Introduction

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 studied this problem in an earlier article (see Schellhorn [10]), where we explained one application of this result to the field of Monte Carlo simulation. We compare our exact results to the approximation given in Schellhorn [10]

Model and results

Formulae for change of Gaussian measures

Example

Conclusions