Minimal fq-martingale measures for exponential levy processes

Let L be a multidimensional Levy process under P in its own filtration. The p-minimal martingale measure Q(q) is defined as that equivalent local martingale measure for 8 (L) which minimizes the f(q)-divergence E[(dQ/dp)(q)] for fixed q epsilon (-infinity, 0) boolean OR (1, infinity). 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 SE arrow 1 in entropy to the minimal entropy martingale measure.