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 $(\Omega ,\mathcal{F},\mathbb{P},({\mathcal{F}}_t{)}_{t\ge 0})$ satisfying some technical conditions, a $\sigma$-finite measure $\mathcal{Q}$ on $(\Omega ,\mathcal{F})$, such that for all $t\ge 0$, and for all events ${\Lambda }_t\in {\mathcal{F}}_t$:

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 $\mathbb{P}$ with respect to a large class of functionals. The measure $\mathcal{Q}$ appears to be the unifying object in these problems.