{
  "id": "2108.11984",
  "version": "v1",
  "published": "2021-08-26T18:27:57.000Z",
  "updated": "2021-08-26T18:27:57.000Z",
  "title": "Characterization of a new class of stochastic processes including all known extensions of the class $(Σ)$",
  "authors": [
    "Fulgence Eyi Obiang",
    "Paule Joyce Mbenangoye",
    "Octave Moutsinga"
  ],
  "comment": "17 Pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "This paper contributes to the study of class $(\\Sigma^{r})$ as well as the c\\`adl\\`ag semi-martingales of class $(\\Sigma)$, whose finite variational part is c\\`adl\\`ag instead of continuous. The two above-mentioned classes of stochastic processes are extensions of the family of c\\`adl\\`ag 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}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-26T18:27:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic processes",
      "finite variational part",
      "second characterization result",
      "mentioned characterization result",
      "nikeghbali"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}