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

Bingzhen Genga, Shijie Wanga, Yang Yang

TL;DR

The paper develops second-order tail expansions for discounted aggregate claims in continuous-time renewal risk models with a constant interest force, addressing both without-by-claims and with-by-claims frameworks. Leveraging second-order subexponentiality and a weighted Kesten-type inequality for randomly weighted sums, it derives uniformly accurate second-order asymptotics, extending prior first-order results. For densities in regularly varying classes, explicit second-order terms are obtained, and simulations confirm the improved accuracy over first-order asymptotics. The findings enhance risk evaluation for heavy-tailed claim sizes in actuarial settings and provide practical tools for refined ruin probability approximations under discounting.

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.

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

TL;DR

The paper develops second-order tail expansions for discounted aggregate claims in continuous-time renewal risk models with a constant interest force, addressing both without-by-claims and with-by-claims frameworks. Leveraging second-order subexponentiality and a weighted Kesten-type inequality for randomly weighted sums, it derives uniformly accurate second-order asymptotics, extending prior first-order results. For densities in regularly varying classes, explicit second-order terms are obtained, and simulations confirm the improved accuracy over first-order asymptotics. The findings enhance risk evaluation for heavy-tailed claim sizes in actuarial settings and provide practical tools for refined ruin probability approximations under discounting.

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.
Paper Structure (6 sections, 5 theorems, 48 equations, 1 figure)

This paper contains 6 sections, 5 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Second order asymptotic for continuous-time renewal risk models without by-claims
  3. Second order asymptotic for continuous-time renewal risk models with by-claims
  4. Simulation
  5. Conclusion
  6. Appendix

Key Result

Theorem 2.1

Consider the discounted aggregate claims described in eq:Dr. If $F\in\widetilde{{\mathscr S}_2}$, then, for arbitrary fixed $T\in\Lambda$, it holds uniformly for all $t \in \Lambda_T$ that

Figures (1)

  • Figure :

Theorems & Definitions (13)

  • Definition 1.1
  • Definition 1.2
  • Theorem 2.1
  • Remark 2.1
  • Remark 2.2
  • Definition 2.1
  • Corollary 2.1
  • Theorem 3.1
  • Remark 3.1
  • Remark 3.2
  • ...and 3 more