On a Bivariate Poisson Negative Binomial Risk Process

Abstract

In this paper we define a bivariate counting process as a compound Poisson process with bivariate negative binomial compounding distribution. We investigate some of its basic properties, recursion formulas and probability mass function. Then we consider a risk model in which the claim counting process is the defined bivariate Poisson negative binomial process. For the defined risk model we derive the distribution of the time to ruin in two cases and the corresponding Laplace transforms. We discuss in detail the particular case of exponentially distributed claims.