Indifference price with unbounded processes and claims: an Orlicz space approach

We consider a financial market where the discounted prices of the assets available for trading are modeled by a semimartingale that is 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. We study the utility maximization problem with an unbounded random endowment in this general context. Applying the theory of Orlicz spaces, this problem is stated and solved in a unified framework for both increasing concave utility functions: u:R->R and u:(0, infinity)->R. We then apply the duality relation to compute the indifference price of a claim satisfying weak integrability conditions. For the exponential utility function, the indifference price leads to a convex risk measure whose dual representation is based on a set of singular functionals belonging to an appropriate Orlicz space's dual space. The penalty term is split into an entropic component and a singular one interpreted as a measure of catastrophic events.

(*) The talk is based on joint works with S. Biagini and S. Biagini, M. Grasselli, and T. Hurd.