Minimal $f^q$-martingale measures for exponential L\'evy processes

Let $L$ be a multidimensional L'evy 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 $E[(dQ\mathrm{/}dP{)}^q]$ for fixed $q\in (-\mathrm{\infty },0)\cup (1,\mathrm{\infty })$. 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 $q\searrow 1$ in entropy to the minimal entropy martingale measure.