Explicit solution of relative entropy weighted control

TL;DR

This paper provides a comprehensive analysis of relative entropy weighted control problems, characterizing optimal solutions and deriving explicit formulas, especially for Wiener processes, with applications to Brownian motion maxima.

Contribution

It offers a complete characterization of when finite optimal values and minimizing measures exist, and derives explicit solutions involving Malliavin derivatives for Wiener measure cases.

Findings

Explicit optimal drift for Brownian motion maximum case

Characterization of finite relative entropy changes in Wiener measure

Monte-Carlo approximation for optimal drift

Abstract

We consider the minimization over probability measures of the expected value of a random variable, regularized by relative entropy with respect to a given probability distribution. In the general setting we provide a complete characterization of the situations in which a finite optimal value exists and the situations in which a minimizing probability distribution exists. Specializing to the case where the underlying probability distribution is Wiener measure, we characterize finite relative entropy changes of measure in terms of square integrability of the corresponding change of drift. For the optimal change of measure for the relative entropy weighted optimization, an expression involving the Malliavin derivative of the cost random variable is derived. The theory is illustrated by its application to several examples, including the case where the cost variable is the maximum of a standard Brownian motion over a finite time horizon. For this example we obtain an exact optimal drift, as well as an approximation of the optimal drift through a Monte-Carlo algorithm.