Propagation of Convexity by Markovian and Martingalian Semigroups

摘要

We study the propagation of convexity by positive Markovian semigroups Q_t on R_~* which are also Martingalian (e.i. Q_t 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 os such semigroups, and even a Levy-Kintchine type decomposition. We give some applications to the s.d.e. dS_t = #sigma#(S_t)S_t dB_t with B standard brownian motion.