On the minimal entropy martingale measure

Abstract

Let X be a locally bounded semimartingale. Using the theory of \textit{BMO}-martingales we give a sufficient criterion for a martingale measure for X to minimize relative entropy among all martingale measures. This is applied to prove convergence of the q-optimal martingale measure to the minimal entropy martingale measure in entropy for $q\downarrow 1$ under the assumption that X is continuous and that the density process of some equivalent martingale measure satisfies a reverse $\mathit{LLogL}$\small -inequality.