Uniformly asymptotic behavior for the tail probability of discounted aggregate claims in the time-dependent risk model with upper tail asymptotically independent claims

ABSTRACT

In this paper, we consider the tail behavior of discounted aggregate claims in a dependent risk model with constant interest force, in which the claim sizes are of upper tail asymptotic independence structure, and the claim size and its corresponding inter-claim time satisfy a certain dependence structure described by a conditional tail probability of the claim size given the inter-claim time before the claim occurs. For the case that the claim size distribution belongs to the intersection of long-tailed distribution class and dominant variation class, we obtain an asymptotic formula, which holds uniformly for all times in a finite interval. Moreover, we prove that if the claim size distribution belongs to the consistent variation class, the formula holds uniformly for all times in an infinite interval.

Keywords:

• Discounted aggregate claims
• Time-dependence
• Uniform asymptotics
• Upper tail asymptotically independence

Mathematics Subject Classification:

• 62P05
• 62E20

Acknowledgments

The authors would like to thank the anonymous referee for her/his constructive comments and suggestions, which greatly improved the quality and the presentation of this paper.

Funding

The paper was supported by the National Natural Science Foundation of China (No. 11501295), the Postdoctoral Science Foundation of China (No. 2015M580415), the Natural Science Foundation of Jiangsu Province of China (No. BK20151459), the Postdoctoral Science Foundation of Jiangsu Province of China (No. 1501004B), the Natural Science Foundation of Jiangsu Higher Education Institutions of China (Nos. 13KJB110014 and 14JB110014) and the Projects Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and the Key Lab of Financial Engineering of Jiangsu Province.