Characterization of a New Class of Stochastic Processes Including all Known Extensions of the Class (\(\Sigma\))

Abstract

The class (\(\Sigma\)) is an important family of semimartingales defined by Yor. These processes play a key role in the theory of probability and their applications. For instance, such processes are used to resolve the Skorokhod Imbedding Problem and to construct solutions for homogeneous and inhomogeneous skew Brownian Motion equations. This paper contributes to the study of classes (\(\Sigma\)) and (\(\Sigma^r\)). But, instead of considering as it is customary, the semi-martingales whose finite variational part is continuous, we will consider those whose finite variational part is càdlàg. The two main contributions of this paper are as follows. First, we present a new characterization result for the stochastic processes of class (\(\Sigma^r\)). Second, we provide a framework for unifying the studies of classes (\(\Sigma\)) and (\(\Sigma^r\)). More precisely, we define and study a new larger class that we call class (\(\Sigma^g\)) and for which we give characterization results. In addition, we derive some structural properties inspired of those obtained for classes (\(\Sigma\)) and (\(\Sigma^r\)). Finally, we show that some processes of this new class can take the form of relative martingales. More precisely, we derive a formula allowing to recover some processes of the class (\(\Sigma^g\)) from an honest time and their final value.