Before revision: We consider the exponential Levy models and we study the conditions under which f-minimal equivalent martingale measure preserves Levy property. Then we give a general formula for optimal strategy in a sense of utility maximization. Finally, we study change-point exponential Levy models, namely we give the conditions for the existence of f-minimal equivalent martingale measure and we obtain a general formula for optimal strategy from point of view of the utility maximization .
After revision: We consider the exponential Levy models with change-point which is a random variable, independent from initial Levy processes. On canonical space with initially enlarged filtration we describe all equivalent martingale measures for change-point exponential Levy models, namely we give the conditions for the existence of f-minimal equivalent martingale measure and we obtain a general formula for optimal strategy from point of view of the utility maximization .