# On Some Properties of Universal Sigma-Finite Measures Associated with a Remarkable Class of Submartingales

### Joseph Najnudel

Universität Zürich, Switzerland### Ashkan Nikeghbali

Universität Zürich, Switzerland

## Abstract

In a previous work, we associated with any submartingale $X$ of class $(\Sigma)$, defined on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$ satisfying some technical conditions, a $\sigma$-finite measure $\mathcal{Q}$ on $(\Omega, \mathcal{F})$, such that for all $t \geq 0$, 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.

## Cite this article

Joseph Najnudel, Ashkan Nikeghbali, On Some Properties of Universal Sigma-Finite Measures Associated with a Remarkable Class of Submartingales. Publ. Res. Inst. Math. Sci. 47 (2011), no. 4, pp. 911–936

DOI 10.2977/PRIMS/56