Robust projections in the class of martingale measures

Abstract

Given a convex function $f$ and a set $\Q$ of probability measures, we consider the problem of minimizing the robust $f$-divergence $\infq f(P|Q)$ over the class $\PP$ of martingale measures. Under mild conditions on $\PP$ and $\Q$ we show that a minimizer exists within the class $\PP$ if $\lim_{x \rightarrow \infty} f(x)/x = \infty$. If $\lim_{x \rightarrow \infty} f(x)/x = 0$ then there is a minimizer in a class $\bar\PP$ of extended martingale measures defined on the predictable $\sigma$-field. We also explain how both cases are connected to recent developments in the theory of optimal portfolio choice, in particular to robust extensions of the classical expected utility criterion.