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

• Published in: arXiv.org
• Main Authors: Fulgence Eyi Obiang, Paule Joyce Mbenangoye, Moutsinga, Octave
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 26.08.2021
• Subjects: Martingales; Stochastic models; Stochastic processes; Theorems

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}.