Paper: arXiv 2501.02545

Authors: Bingzhen Genga, Shijie Wanga, Yang Yang

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.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 2.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous mathematical proofs for second-order asymptotic expansions in continuous-time renewal risk models with second-order subexponential claims. However, it is primarily a theoretical actuarial probability paper with minimal empirical or backtesting content beyond simple illustrative numerical simulations.

Research Flowchart

  flowchart TD
    A[Goal: Derive second-order asymptotic tail probabilities for discounted aggregate claims] --> B[Inputs: Continuous-Time Renewal Risk Models<br><i>with & without by-claims</i>]
    B --> C[Construct Asymptotic Theory &<br>Weighted Kesten-Type Inequality]
    C --> D[Analyze Randomly Weighted Sums for<br>Second-Order Subexponential RVs]
    D --> E[Perform Numerical Validation Studies]
    E --> F[Outcome: Highly Precise Second-Order<br>Asymptotic Formulae Outperforming First-Order Models]

Cited References (37)

  1. Investigation a Dependent Generalized Compound Renewal Risk Process Involving the Uniformly Bounded Copula Function (2024) — link
  2. Asymptotics for a Bidimensional Renewal Risk Model with Subexponential Main Claims and Delayed Claims (2023) — link
  3. Second-Order Tail Behavior for Stochastic Discounted Value of Aggregate Net Losses in a Discrete-Time Risk Model (2022) — link
  4. Asymptotic Finite-Time Ruin Probabilities for a Bidimensional Delay-Claim Risk Model with Subexponential Claims (2022) — link
  5. Asymptotics for a time-dependent renewal risk model with subexponential main claims and delayed claims (2021) — link
  6. Second Order Asymptotics for Infinite-Time Ruin Probability in a Compound Renewal Risk Model (2021) — link
  7. Second order tail behaviour of randomly weighted heavy-tailed sums and their maxima (2020) — link
  8. Second order tail approximation for the maxima of randomly weighted sums with applications to ruin theory and numerical examples (2019) — link
and 29 more
  1. Second order asymptotics for ruin probabilities of the delayed renewal risk model with heavy-tailed claims (2019) — link
  2. On asymptotic finite-time ruin probability of a renewal risk model with subexponential main claims and delayed claims (2019) — link
  3. Asymptotics for a delay-claim risk model with diffusion, dependence structures and constant force of interest (2019) — link
  4. Second order tail behaviour for heavy-tailed sums and their maxima with applications to ruin theory (2014) — link
  5. On pairwise quasi-asymptotically independent random variables and their applications (2013) — link
  6. Second order asymptotics for ruin probabilities in a renewal risk model with heavy-tailed claims (2012) — link
  7. An Introduction to Heavy-Tailed and Subexponential Distributions (2011) — link
  8. Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model (2010) — link
  9. Extremes on the discounted aggregate claims in a time dependent risk model (2010) — link
  10. A uniform asymptotic estimate for discounted aggregate claims with subexponential tails (2008) — link
  11. Heavy Tails of Discounted Aggregate Claims in the Continuous-Time Renewal Model (2007) — link
  12. The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims (2007) — link
  13. THE FINITE-TIME RUIN PROBABILITY OF THE COMPOUND POISSON MODEL WITH CONSTANT INTEREST FORCE (2005) — link
  14. On ultimate ruin in a delayed-claims risk model (2005) — link
  15. Asymptotics for Sums of Random Variables with Local Subexponential Behaviour (2003) — link
  16. Modelling Extremal Events for Insurance and Finance (2002) — link
  17. Ruin probabilities for time-correlated claims in the compound binomial model (2001) — link
  18. Ruin estimates under interest force (1995) — link
  19. Second order subexponential distributions (1991) — link
  20. Subexponential distributions and integrated tails (1988) — link
  21. Second order behaviour of the tail of a subordinated probability distribution (1986) — link
  22. Ruin probabilities allowing for delay in claims settlement (1985) — link
  23. A Note on Cumulative Sums (1946) — link
  24. “On a discrete time risk model with time-delayed claims and a constant dividend barrier” (2019)
  25. Extreme value theory : an introduction (2006) — link
  26. Ruin probabilities in the presence of heavy tails and interest rates (1998) — link
  27. Second-Order Regular Variation and Rates of Convergence in Extreme-Value Theory (1996) — link
  28. Subexponential distributions and characterizations of related classes (1989) — link
  29. A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes (1964) — link

0 of 37 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.