Minimal entropy preserves the Lévy property: how and why

Let L be a multidimensional Lévy process under P in its own filtration and consider all probability measures Q turning L into a local martingale. The minimal entropy martingale measure Q E is the unique Q which minimizes the relative entropy with respect to P. We prove that L is still a Lévy process under Q E and explain precisely how and why this preservation of the Lévy property occurs.