The supermartingale property of the optimal wealth process for general semimartingale

We consider a stochastic financial incomplete market where the price processes are described by a vector valued semimartingale that is possibly non locally bounded. We face the classical problem of the utility maximization from terminal wealth, under the assumption that the utility function is finite valued and smooth on the entire real line and satisfies Reasonable Asymptotic Elasticity. In this general setting, it was shown in Biagini and Frittelli (2005) that the optimal claim admits an integral representation as soon as the minimax sigma-martingale measure is equivalent to the reference probability measure. We show that the optimal wealth process is in fact a supermartingale with respect to every sigma-martingale measure with finite generalized entropy, thus extending the analogous result proved by Schachermayer (2003) in the locally bounded case.