A Generalization of Doob’s Maximal Identity

In this paper, using martingale techniques, we prove a generalization of Doob’s maximal identity in the setting of continuous nonnegative local submartingales (Xt) of the form: Xt = Nt + At, where the measure (dAt) is carried by the set {t : Xt = 0}. In particular, we give a multiplicative decomposition for the Azéma supermartingale associated with some last passage times related to such processes and we prove that these non-stopping times contain very useful information. As a consequence, we obtain the law of the maximum of a continuous nonnegative local martingale (Mt) which satisfies M∞ = ψ(supt≥0Mt) for some measurable function ψ as well as the law of the last time this maximum is reached.