## Abstract

In this paper, for any submartingale of class (Sigma) defined on a filtered probability space (Omega, F, P, (F-t)(t >= 0)) satisfying some technical conditions, we associate a sigma-finite measure Q on (Omega, F), such that for all t >= 0, and for all events Lambda(t) is an element of F-t:

Q[Lambda(t), g <= t ] = Ep[1(Lambda 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. (c) 2012 Elsevier B.V. All rights reserved.