On some properties of universal sigma-finite measures associated with a remarkable class of submartingales

Abstract

In a previous work, we associated with any submartingale X of class (Sigma), defined on a filtered probability space (Omega, F, (F-t)(t >= 0), P) satisfying some technical conditions, a sigma-finite measure Q on (Omega, F) such that for all t >= 0, and for all events Lambda(t) is an element of F-t, Q[Λt,g≤t]=EP[1ΛtXt], where g is the last hitting time of zero by the process X. In this paper we establish some remarkable properties of this measure from which we also deduce a universal class of penalisation results of the probability measure P with respect to a large class of functionals. The measure Q appears to be the unifying object in these problems.