Multivariate Tempered Space-Fractional Negative Binomial Process and Risk Models with Shocks

TL;DR

This paper introduces a new multivariate tempered space-fractional negative binomial process, explores its properties, and applies it to risk models with shocks, deriving key risk measures and demonstrating long-range dependence.

Contribution

It defines the MTSFNBP, studies its properties, and applies it to risk modeling, providing new insights into shock-driven risk processes with long-range dependence.

Findings

Derived the Le9vy measure density for MTSFNBP.

Established the stochastic equivalence to a generalized Cramer-Lundberg model.

Obtained ruin probabilities and related risk measures.

Abstract

In this paper, we first define the multivariate tempered space-fractional Poisson process (MTSFPP) by time-changing the multivariate Poisson process with an independent tempered {\alpha}-stable subordinator. Its distributional properties, the mixture tempered time and space variants and their PDEs connections are studied. Then we define the multivariate tempered space-fractional negative binomial process (MTSFNBP) and explore its key features. The L\'evy measure density for the MTSFNBP is also derived. We present a bivariate risk model with a common shock driven by the tempered space-fractional negative binomial process. We demonstrate that the total claim amount process is stochastically equivalent to a univariate generalized Cramer-Lundberg risk model. In addition, some important ruin measures such as ruin probability, joint distribution of time to ruin and deficit at ruin along with governing integro-differential equations are obtained. Finally we show that the underlying risk process exhibits the long range dependence property.