Analysis of a MAP Risk Model with Stochastic Incomes, Inter-Dependent Phase-Type Claims and a Constant Barrier

Abstract

Inspired by the problems with random income feature, this paper focuses on an insurance risk model with MAP inter-arrival time for premiums as well as claims. We study the model for a convex combination of two types of inter-dependent Phase-type claims, where the probability of claim switching is directly associated with the inter-arrival time of claims. Furthermore, the surplus process of this model is assumed to be restricted by a horizontal barrier “b” above the initial surplus “u”. The transient analysis of the corresponding Markovian fluid flow model is considered to develop the integral equations governing the Gerber–Shiu function and the expected discounted dividends paid until ruin. The closed-form solutions for these integral equations are obtained in terms of Lundberg roots. When the premium sizes are Phase-type distributed, the solutions are explicit at “u = b”. For “u ≤ b”, the solutions are explicit when the premium sizes are distributed exponentially. Finally, to validate and present the tractability of these solution expressions, some numerical illustrations are provided in individual cases.