Minimal $f^q$-martingale measures for exponential Lévy processes

Details Minimal $f^q$-martingale measures for exponential Lévy processes Minimal $f^q$-martingale measures for exponential Lévy processes

2007-10-30

arxiv.org/abs/0710.5594v1

Annals of Applied Probability 2007, Vol. 17, No. 5,6, 1615-1638

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AAP439 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/07-AAP439

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 $E[(dQ/dP)^q]$ for fixed $q\in(-\infty,0)\cup(1,\infty)$. 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 $q\searrow1$ in entropy to the minimal entropy martingale measure.

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