Second order asymptotics for discounted aggregate claims of continuous-time renewal risk models with constant interest force

TL;DR

This paper develops second order asymptotic formulas for tail probabilities of discounted aggregate claims in continuous-time renewal risk models, enhancing precision over first order approximations through new theoretical tools.

Contribution

It introduces second order asymptotic expansions for these risk models using novel weighted Kesten-type inequalities for subexponential variables.

Findings

More accurate tail probability approximations than first order formulas

Theoretical development of weighted Kesten-type inequalities

Numerical studies demonstrating improved precision

Abstract

This paper investigates the second order asymptotic expansion for tail probabilities of discounted aggregate claims in continuous-time renewal risk models with constant interest force. Concretely, two types of continuous-time renewal risk models without and with by-claims are separately discussed. By constructing the asymptotic theory and weighted Kesten-type inequality of randomly weighted sums for second order subexponential random variables, second order asymptotic formulae for these two risk models are firstly built. In comparison of the first order asymptotic formulae, our results are more superior and precise, which are demonstrated by some simple numerical studies.