Levy preservation and associated properties for \(f\)-divergence minimal equivalent martingale measures

Abstract

We study such important properties of \(f\)-divergence minimal martingale measure as Levy preservation property, scaling property, invariance in time property for exponential Levy models. We give some useful decomposition for \(f\)-divergence minimal martingale measures and we answer on the question which form should have \(f\) to ensure mentioned properties. We show that \(f\) is not necessarily common \(f\)-divergence. For common \(f\)-divergences, i.e. functions verifying \(f"(x) = ax^ {\gamma},\, a>0,\, \gamma \in \mathbb R\), we give necessary and sufficient conditions for existence of \(f\)-minimal martingale measure.