Robust utility maximization for L\'evy processes: Penalization and solvability

In this paper the robust utility maximization problem for a market model based on L\'evy processes is analyzed. The interplay between the form of the utility function and the penalization function required to have a well posed problem is studied, and for a large class of utility functions it is proved that the dual problem is solvable as well as the existence of optimal solutions. The class of equivalent local martingale measures is characterized in terms of the parameters of the price process, and the connection with convex risk measures is also presented.