A remarkable $\sigma$-finite measure associated with last passage times and penalisation problems

TL;DR

This paper explores a special class of submartingales called $(\Sigma)$, their connection to a universal sigma-finite measure, and applications to penalization problems on path spaces, providing a comprehensive overview of these mathematical structures.

Contribution

It introduces a global perspective on $(\Sigma)$ submartingales and their relationship with a universal sigma-finite measure and penalization techniques.

Findings

Characterization of $(\Sigma)$ submartingales

Establishment of a universal sigma-finite measure

Applications to penalization problems on path spaces

Abstract

In this paper, we give a global view of the results we have obtained in relation with a remarkable class of submartingales, called $(\Sigma)$, and its links with a universal sigma-finite measure and penalization problems on the space of continuous and cadlag paths.