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 . We illustrate our results considering  Black-Scholes model with change-point.
We study 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 model and we give the conditions for the existence of f-minimal equivalent martingale measure . Using the connection between utility maximisation and f-divergence minimisation, we obtain a general formula for optimal strategy in change-point case for initially enlarged filtration and also for progressively enlarged filtration when the utility is exponential . We illustrate our results considering the Black-Scholes model with change-point.
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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
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exponential Levy models, namely
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