Characterization of optional Submartingales of a new class (Σl)

Resumo

In this paper, we shall define a new class denote by (Σ^l) of optional submartingales of the form X_t = M_t + A_t, where (M_t)_t≥0 is a is a càdlàg (right continuous with left limits) local martingale, (A_t)_t≥0 is a làdlàg increasing process, which can be decomposed as A_t = A_t^d + A_t^g  such that A^d is a càdlàg increasing
process, A^g is a càglàd increasing process, the measure (dA^d) is carried by the set {t : X_t-= 0}, and the measure (dA^g₊₎ is carried by the set {t : X_t = 0}. Our main purpose in this work is to study the positive and negative parts of these processes, and establish some martingale characterizations, then show the formula
of relative martingales in terms of last passage times, finally calculate a predictable compensator by using balayage formula.