On some universal -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 , ( F t ) 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 Λ t ∈ F t : Q [ Λ t , g ≤ t ] = E P [ 1 Λ t X t ] , 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.