Characterization of a new class of stochastic processes including all known extensions of the class $(\Sigma )$

Abstract

This paper contributes to the study of class $({\Sigma }^r)$ as well as the càdlàg semi-martingales of class $(\Sigma )$, whose finite variational part is càdlàg instead of continuous. The two above-mentioned classes of stochastic processes are extensions of the family of càdlàg semi-martingales of class $(\Sigma )$ considered by Nikeghbali \cite{nik} and Cheridito et al. \cite{pat}; i.e., they are processes of the class $(\Sigma )$, whose finite variational part is continuous. The two main contributions of this paper are as follows. First, we present a new characterization result for the stochastic processes of class $({\Sigma }^r)$. More precisely, we extend a known characterization result that Nikeghbali established for the non-negative sub-martingales of class $(\Sigma )$, whose finite variational part is continuous (see Theorem 2.4 of \cite{nik}). 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 $({\Sigma }^g)$. In particular, we establish two characterization results for the stochastic processes of the said class. The first one characterizes all the elements of class $({\Sigma }^g)$. Hence, we derive two corollaries based on this result, which provides new ways to characterize classes $(\Sigma )$ and $({\Sigma }^r)$. The second characterization result is, at the same time, an extension of the above mentioned characterization result for class $({\Sigma }^r)$ and of a known characterization result of class $(\Sigma )$ (see Theorem 2 of \cite{fjo}). In addition, we explore and extend the general properties obtained for classes $(\Sigma )$ and $({\Sigma }^r)$ in \cite{nik,pat,mult, Akdim}.

17 Pages