Minimal fq-martingale measures for exponential Lévy processes

Abstract

Let L be a multidimensional Lévy process under P in its own filtration. The f^q-minimal martingale measure Q_q is defined as that equivalent local martingale measure for $\mathcal {E}(L)$ which minimizes the f^q-divergence E[(dQ/dP)^q] for fixed q∈(−∞, 0)∪(1, ∞). We give necessary and sufficient conditions for the existence of Q_q 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 Q_q converges for q↘1 in entropy to the minimal entropy martingale measure.