  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.
