2021/08/26 by Fulgence Eyi Obiang, Obiang, Fulgence Eyi, Paule Joyce Mbenangoye +3
Mathematics · #Approximation Theory and Sequence Spaces #FOS: Mathematics #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2108.11984
openalex publication_date 2021/08/26 · openalex created_date 2022/10/25 · openalex updated_date 2026/07/28
This paper contributes to the study of class (Σr) as well as the càdlàg semi-martingales of class (Σ), 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 (Σ) considered by Nikeghbali \citenik and Cheridito et al. \citepat; i.e., they are processes of the class (Σ), 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 (Σr). More precisely, we extend a known characterization result that Nikeghbali established for the non-negative sub-martingales of class (Σ), whose finite variational part is continuous (see Theorem 2.4 of \citenik). Second, we provide a framework for unifying the studies of classes (Σ) and (Σr). More precisely, we define and study a new larger class that we call class (Σg). In particular, we establish two characterization results for the stochastic processes of the said class. The first one characterizes all the elements of class (Σg). Hence, we derive two corollaries based on this result, which provides new ways to characterize classes (Σ) and (Σr). The second characterization result is, at the same time, an extension of the above mentioned characterization result for class (Σr) and of a known characterization result of class (Σ) (see Theorem 2 of \citefjo). In addition, we explore and extend the general properties obtained for classes (Σ) and (Σr) in \citenik,pat,mult, Akdim.