Pólya–Aeppli of Order k Risk Model

Abstract

In this study, we define the Pólya–Aeppli process of order k as a compound Poisson process with truncated geometric compounding distribution with success probability 1 − ρ > 0 and investigate some of its basic properties. Using simulation, we provide a comparison between the sample paths of the Pólya–Aeppli process of order k and the Poisson process. Also, we consider a risk model in which the claim counting process {N(t)} is a Pólya-Aeppli process of order k, and call it a Pólya—Aeppli of order k risk model. For the Pólya–Aeppli of order k risk model, we derive the ruin probability and the distribution of the deficit at the time of ruin. We discuss in detail the particular case of exponentially distributed claims and provide simulation results for more general cases.

Keywords:

• Distributions of order k
• Pólya–Aeppli process
• Ruin probability
• Sensitivity analysis
• Simulation

Mathematics Subject Classification:

• 60K10
• 62P05