Expected Utility Maximization in Exponential Levy Models

We consider a standard problem of maximizing expected utility for the terminal value of a portfolio’s capital in the case where the asset price process is a stochastic exponential of the Levy process. Situations where utility has a logarithmic, power and exponential form are studied. The aim of the paper is to in at most possible general assumptions solve the main and the dual problems and obtain answers in rather explicit form in terms of the Levy triplet. Using it in logarithmic case we calculate the conditions which determine if the solution to the dual problem is an equivalent martingale measure, a martingale or a supermartingale.