Uniform Asymptotics for a Delay-Claims Risk Model with Constant Force of Interest and By-Claims Arriving according to a Counting Process

Qingwu Gao,Xijun Liu

doi:10.1155/2019/6385647

Abstract

The insurance risk model involving main claims and by-claims has been traditionally studied under the assumption that every main claim may be accompanied with a by-claim occurring after a period of delay, but in reality each main claim can cause many by-claims arriving according to a counting process. To this end, we construct a new insurance risk model that is also perturbed by diffusion with constant force of interest. In the presence of heavy tails and dependence structures among modelling components, we obtain some asymptotic results for the finite-time ruin probability and the tail probability of discounted aggregate claims, where the results hold uniformly for all times in a finite or infinite interval.

Highlights

X ≥ 0 is the initial capital of an insurance company, r > 0 is the constant force of interest, c ≥ 0 is the constant rate of premium, 1A is the indicator function of an event A, Xi, i ≥ 1, are the insurer’s main claims such that each main claim may cause a by-claim Yi, i ≥ 1, occurring after a delay period Ti, i ≥ 1, and τi, i ≥ 1, are the main-claim arrival times constituting a renewal counting process, N (t) = sup {i ≥ 1 : τi ≤ t}, t ≥ 0, (2)
We propose a new insurance risk model, which involves the main claims and the by-claims such that each main claim can cause multiple by-claims arriving according to a counting process, and in which all the modelling components satisfy the following assumptions
In this paper we consider the new insurance risk model (4) with a feature that each main claim can cause multiple by-claims arriving according to a counting process after a period of delay and investigate the uniform asymptotics for the finite-time ruin probability and the tail probability of the discounted aggregate claims

Summary

Introduction

Consider the classical delay-claims risk model with constant force of interest (see Li [1]), whose wealth process is expressed as t. We propose a new insurance risk model, which involves the main claims and the by-claims such that each main claim can cause multiple by-claims arriving according to a counting process, and in which all the modelling components satisfy the following assumptions. In this paper we consider the new insurance risk model (4) with a feature that each main claim can cause multiple by-claims arriving according to a counting process after a period of delay and investigate the uniform asymptotics for the finite-time ruin probability and the tail probability of the discounted aggregate claims.

Main Results

Some Lemmas

Proofs of Main Results