We consider a financial market where the discounted prices of the assets available for trading are modeled by semimartingales that are not assumed to be locally bounded. In this case the appropriate class of admissible integrands is defined through a random variable W that controls the losses incurred in trading. Applying the theory of Orlicz spaces, and convex analysis we study the utility maximization problem with an unbounded random endowment.

We then apply the duality relation to compute the indifference price of a claim satisfying weak integrability conditions. The indifference price leads to a convex risk measure defined on the Orlicz space associated to the utility function.

The talk is based on joint works with S. Biagini and with S. Biagini, M. Grasselli.