Discrete-Time Risk Models on Time Series for Count Random Variables

In this paper we consider various specifications of the general discrete-time risk model in which a serial dependence structure is introduced between the claim numbers for each period. We consider risk models based on compound distributions considering several examples of discrete variate time series as specific temporal dependence structures: Poisson MA (1) process, Poisson AR (1) process, Markov Bernoulli process and Markov regime-switching process. In these models, we derive expressions for a function that allow us to find the Lundberg coefficient. Specific cases for which an explicit expression can be found for the Lundberg coefficient are also presented. Numerical examples are provided to illustrate different topics discussed in the paper.

Keywords: Discrete-time risk model; Poisson MA (1) process; Poisson AR (1) process; Markov Bernoulli Process; Markovian Environment; Lundberg Coefficient.