From an Itô type calculus for Gaussian processes to integrals of log-normal processes increasing in the convex order

Abstract

We present an Itô type formula for a Gaussian process, in which only the one-marginals of the Gaussian process are involved. Thus, this formula is well adapted to the study of processes increasing in the convex order, in a Gaussian framework. In particular, we give conditions ensuring that processes defined as integrals, with respect to one parameter, of exponentials of two-parameter Gaussian processes, are increasing in the convex order with respect to the other parameter. Finally, we construct Gaussian sheets allowing to exhibit martingales with the same one-marginals as the previously defined processes.