Minimal fq-Martingale measures for exponential lévy processes

Abstract

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 ε(L) which minimizes the fq -divergence E[(dQ/dP)q] for fixed q ∈ (∞, 0) U (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. © Institute of Mathematical Statistics, 2007.