Generalized Fractional Risk Process

TL;DR

This paper introduces a new generalized fractional risk process based on a compound fractional counting process, analyzing its properties, dependence structure, and asymptotic ruin probabilities for different claim size distributions.

Contribution

It defines the CGFCP and GFRP, explores their properties, dependence, and provides asymptotic ruin probability results, extending fractional risk models.

Findings

GFRP exhibits long-range dependence.

GFRP's increment process shows short-range dependence.

Asymptotic ruin probabilities are derived for light and heavy-tailed claims.

Abstract

In this paper, we define a compound generalized fractional counting process (CGFCP) which is a generalization of the compound versions of several well-known fractional counting processes. We obtain its mean, variance, and the fractional differential equation governing the probability law. Motivated by Kumar et al. (2020), we introduce a fractional risk process by considering CGFCP as the surplus process and call it generalized fractional risk process (GFRP). We study the martingale property of the GFRP and show that GFRP and the associated increment process exhibit the long-range dependence (LRD) and the short-range dependence (SRD) property, respectively. We also define an alternative to GFRP, namely AGFRP which is premium wise different from the GFRP. Finally, the asymptotic structure of the ruin probability for the AGFRP is established in case of light-tailed and heavy-tailed claim sizes.