Levy preservation and associated properties for $f$-divergence minimal equivalent martingale measures

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 $f\mathrm{"}(x)=a{x}^{\gamma },a>0,\gamma \in \mathbb{R}$, we give necessary and sufficient conditions for existence of $f$-minimal martingale measure.