A remarkable σ-finite measure associated with last passage times and penalisation problems

Abstract

In this paper, we give a global view of the results we have obtained in relation with a remarkable class of submartingales, called (Σ), and which are stated by the authors in [“On some universal σ-finite measures and some extensions of Doob's optional stopping theorem”, arXiv:0906.1782 (2009); “A new kind of augmentation of filtrations”, arXiv:0910.4959 (2009); “On some properties of a universal sigma-finite measure associated with a remarkable class of submartingales”, arXiv:0911.2571 (2009); and “On penalisation results related with a remarkable class of submartingales”, arXiv:0911.4365 (2009)]. More precisely, we associate to a given submartingale in this class (Σ), defined on a filtered probability space (Ω,ℱ,P,(ℱ t ) t≥0 ), satisfying some technical conditions, a σ-finite measure