Minimal fq-martingale measures for exponential Lévy processes

Let L be a multidimensional Lévy process under P in its own filtration. The fq-minimal martingale measure Qq is defined as that equivalent local martingale measure for E(L) which minimizes the fq-divergence E[(dQ/dP)q] for fixed q.(.., 0).(1, .). We give necessary and sufficient conditions for the existence of Qq and an explicit formula for its density. For q=2, we relate the sufficient conditions to the structure condition and discuss when the former are also necessary. Moreover, we show that Qq converges for q.1 in entropy to the minimal entropy martingale measure.