The supermartingale property of the optimal wealth process for general semimartingales

We consider an incomplete stochastic financial market where the price processes are described by a vector valued semimartingale that is possibly non locally bounded. We face the classical problem of 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 (Financ. Stoch. 9, 493–517, 2005) that the optimal claim admits an integral representation as soon as the minimax σ-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 σ-martingale measure with finite generalized entropy, thus extending the analogous result proved by Schachermayer (Financ. Stoch. 4, 433–457, 2003) for the locally bounded case.