Propagation of Convexity by Markovian and Martingalian Semigroups

We study the propagation of convexity by positive Markovian semigroups Qt on R^*₊ which are also Martingalian (i.e. Qt Id = Id). This question is related to the management of the volatility risk in theoretical finance. We exhibit a new duality between Markovian semigroups which is an instance of T. Liggett's h-duality. In the continuous case we give a characterization theorem of the infinitesimal generators of such semigroups, and even a Lévy–Kintchine type decomposition. We give some applications to the s.d.e. dSit = ( StdBt with B standard brownian motion.