# 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 $(Σ)$, defined on a filtered probability space $(Ω,F,P,(F_{t})_{t≥0})$ satisfying some technical conditions, a $σ$-finite measure $Q$ on $(Ω,F)$, such that for all $t≥0$, and for all events $Λ_{t}∈F_{t}$:

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 $P$ with respect to a large class of functionals. The measure $Q$ 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