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

TL;DR

This paper investigates the robust utility maximization problem in markets modeled by Lévy processes, focusing on the relationship between utility functions, penalization, and the existence of optimal solutions.

Contribution

It characterizes the conditions for well-posedness and solvability of the dual problem in Lévy process-based markets, linking utility, penalization, and risk measures.

Findings

Dual problem is solvable for a large class of utility functions.

Existence of optimal solutions is established under certain conditions.

Characterization of equivalent local martingale measures in terms of process parameters.

Abstract

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.