Convex comparison inequalities for exponential jump-diffusion processes

Given (Mt)t2R+ and (M t )t2R+ respectively a forward and a backward exponential martingale with jumps and a continuous part, we prove that E( (MtM t )) is non-increasing in t when is a convex function, pro- vided the local characteristics of the stochastic logarithms of (Mt)t2R+ and of (M t )t2R+ satisfy some comparison inequalities. As an application, we de- duce bounds on option prices in markets with jumps, in which the underlying processes need not be Markovian. In this setting the classical propagation of convexity assumption for Markov semigroups (4) is not needed.