An Uncertain Alternating Renewal Insurance Risk Model

Jia Zhai,Haitao Zheng,Manying Bai,Yunyun Jiang

doi:10.1155/2020/3856323

Abstract

The claim process in an insurance risk model with uncertainty is traditionally described by an uncertain renewal reward process. However, the claim process actually includes two processes, which are called the report process and the payment process, respectively. An alternative way is to describe the claim process by an uncertain alternating renewal reward process. Therefore, this paper proposes an insurance risk model under uncertain measure in which the claim process is supposed to be an alternating renewal reward process and the premium process is regarded as a renewal reward process. Then, the paper also gives the inverse uncertainty distribution of the insurance risk process. The expression of ruin index and the uncertainty distribution of the ruin time are derived which both have explicit expressions based on given uncertainty distributions. Finally, several examples are provided to illustrate the modeling ideas.

Highlights

In the above studies, the claim process is regarded as a renewal process or a renewal reward process
The claim process should include two processes: the report process and the payment process. e report process refers to the insured formally notifying the insurance company about an event. e payment process refers to the process whereby the insurer reviews the claim and sees whether the event or situation falls within the risks covered by the policy. at is to say, the insurer will need to determine that the claim meets the terms and conditions of the insurance policy
Inspired by the ideas we have reviewed, this paper proposes an insurance risk model in which the claims follow an uncertain alternating renewal reward process, while the premiums follow an uncertain renewal reward process. e inverse uncertainty distribution of insurance risk process, ruin index, and ruin time are derived

Summary

Research Article An Uncertain Alternating Renewal Insurance Risk Model

E claim process in an insurance risk model with uncertainty is traditionally described by an uncertain renewal reward process. Liu [32] provides an uncertain insurance risk model by applying the renewal reward process and derives the ruin index. In the uncertain insurance risk model, Liu assumes that the premium is a real function proportional to time and the claim amount obeys an uncertain renewal reward process. Few studies have considered the claim process in an insurance risk model as an uncertain alternating renewal process. Inspired by the ideas we have reviewed, this paper proposes an insurance risk model in which the claims follow an uncertain alternating renewal reward process, while the premiums follow an uncertain renewal reward process. ∩ I i 1 􏽮􏼐ξ1i < μ−11(1 − α)􏼑 ∩ 􏼐Pi > Φ− 1(α)􏼑 ∩ 􏼐ξ2i + η2i > μ−21(α) + λ−21(α)􏼑 ∩ 􏼐Ci < Ψ− 1(1 − α)􏼑􏽯

According to the independence of these uncertain variables again that

Ruin index