Multiplicative Decompositions and Frequency of Vanishing of Nonnegative Submartingales

Abstract

Abstract. In this paper, we establish a multiplicative decomposition formula for nonnegative local martingales and use it to characterize the set of continuous local submartingales Y of the form Y = N + A, where the measure dA is carried by the set of zeros of Y . In particular, we shall see that in the set of all local submartingales with the same martingale part in the multiplicative decomposition, these submartingales are the smallest ones. We also study some integrability questions in the multiplicative decomposition and interpret the notion of saturated sets in the light of our results.

On the Study of Processes of $$\sum (H)$$ ∑ ( H ) and $$\sum _\mathrm{s}(H)$$ ∑ s ( H ) Classes

“…Furthermore, C = C ·+g is a continuous and nondecreasing process which is adapted with respect to (F t+g ) t≥0 , with, C 0 = 1. Then, according to Lemma 2.2 of [11], one gets that:…”

“…A large class containing several stochastic processes satisfying (1.1) is the class . The class has been introduced by Yor [13] and studied in a series of articles [3,[6][7][8]10,11] and [9].…”

“…The interest of the present work lies in results established in [11] and [3]. In [11], the multiplicative decomposition formulas have been given for submartingales of class.…”

“…In [11], the multiplicative decomposition formulas have been given for submartingales of class. In [3], some representation results allowing to recover any process of class from its final value and the last time it visited the origin have been established.…”

An essay on the general theory of stochastic processes

“…However, we mention some of them and give a nice application. The reader should refer to [36,55,3,37,65] for more details. We will also give a very elegant application of multiplicative decompositions to the theory of enlargements of filtrations in Section 8.…”