Indifference price with general semimartingales

For utility functions $u$ finite valued on $\mathbb{R}$, we prove a duality formula for utility maximization with random endowment in general semimartingale incomplete markets. The main novelty of the paper is that possibly non locally bounded semimartingale price processes are allowed. Following Biagini and Frittelli \cite{BiaFri06}, the analysis is based on the duality between the Orlicz spaces $(L^{\widehat{u}}, (L^{\widehat{u}})^*)$ naturally associated to the utility function. This formulation enables several key properties of the indifference price $\pi(B)$ of a claim $B$ satisfying conditions weaker than those assumed in literature. In particular, the indifference price functional $\pi$ turns out to be, apart from a sign, a convex risk measure on the Orlicz space $L^{\widehat{u}}$.