A Class of Remarkable Submartingales

In this paper, we consider the special class of positive local submartingales (Xt) of the form: Xt = Nt + At, where the measure (dAt) is carried by the set {t : Xt = 0}. We show that many examples of stochastic processes studied in the literature are in this class and propose a unified approach based on martingale techniques to study them. In particular, we establish some martingale characterizations for these processes and compute explicitly some distributions involving the pair (Xt, At). We also associate with X a solution to the Skorokhod’s stopping problem for probability measures on the positive half-line.