Minimal Entropy Martingale Measure for Lévy Processes

Let X be a real-valued Lévy process under P in its natural filtration. The minimal entropy martingale measure is defined as an absolutely continuous martingale measure that minimizes the relative entropy with respect to P. We show in this paper that the sufficient conditions for its existence, known in literature, are also necessary and give an explicit formula for the infimum of the relative entropy.