On some universal σ-finite measures related to a remarkable class of submartingales

In this paper, for any submartingale of class (Σ) defined on a filtered probability space (Ω,F,P,(Ft)t≥0) satisfying some technical conditions, we associate a σ-finite measure Q on (Ω,F), such that for all t≥0, and for all events Λt∈Ft: Q[Λt,g≤t]=EP[1ΛtXt], where g is the last time for which the process X hits zero. The existence of Q has already been proven in several particular cases, some of them are related with Brownian penalization, and others are involved with problems in mathematical finance. More precisely, the existence of Q in the general case gives an answer to a problem stated by Madan, Roynette and Yor, in a paper about the link between the Black–Scholes formula and the last passage times of some particular submartingales. Moreover, the equality defining Q still holds if the fixed time t is replaced by any bounded stopping time. This generalization can be considered as an extension of Doob’s optional stopping theorem.