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Some continuity estimates for ruin probability and other ruin-related quantities

Lazaros Kanellopoulos

TL;DR

This work addresses the problem of continuity for ruin-related quantities in the classical risk model and in a diffusion-perturbed surplus framework. It advances a contractive-operator approach on weighted function spaces and employs Banach fixed-point arguments to derive explicit continuity inequalities for the ruin probability $\psi(u)$ and the deficit at ruin $G(u,y)$, with bounds expressed via moments and probability metrics such as the Kantorovich distance. In the diffusion-perturbed setting, the ruin probability is linked to a convolution involving a compound geometric kernel $K(u)$, and the authors establish a continuity bound for $\overline{K}(u)$ and propose a BFPT-based iterative scheme to approximate $K(u)$, including concrete numerical illustrations. Collectively, the results provide computable, principled sensitivity bounds for solvency-related metrics and practical iterative methods for complex ruin quantities, enhancing both theoretical insight and numerical usability in actuarial risk analysis.

Abstract

In this paper we investigate continuity properties for ruin probability in the classical risk model. Properties of contractive integral operators are used to derive continuity estimates for the deficit at ruin. These results are also applied to obtain desired continuity inequalities in the setting of continuous time surplus process perturbed by diffusion. In this framework, the ruin probability can be expressed as the convolution of a compound geometric distribution $K$ with a diffusion term. A continuity inequality for $K$ is derived and an iterative approximation for this ruin-related quantity is proposed. The results are illustrated by numerical examples.

Some continuity estimates for ruin probability and other ruin-related quantities

TL;DR

This work addresses the problem of continuity for ruin-related quantities in the classical risk model and in a diffusion-perturbed surplus framework. It advances a contractive-operator approach on weighted function spaces and employs Banach fixed-point arguments to derive explicit continuity inequalities for the ruin probability and the deficit at ruin , with bounds expressed via moments and probability metrics such as the Kantorovich distance. In the diffusion-perturbed setting, the ruin probability is linked to a convolution involving a compound geometric kernel , and the authors establish a continuity bound for and propose a BFPT-based iterative scheme to approximate , including concrete numerical illustrations. Collectively, the results provide computable, principled sensitivity bounds for solvency-related metrics and practical iterative methods for complex ruin quantities, enhancing both theoretical insight and numerical usability in actuarial risk analysis.

Abstract

In this paper we investigate continuity properties for ruin probability in the classical risk model. Properties of contractive integral operators are used to derive continuity estimates for the deficit at ruin. These results are also applied to obtain desired continuity inequalities in the setting of continuous time surplus process perturbed by diffusion. In this framework, the ruin probability can be expressed as the convolution of a compound geometric distribution with a diffusion term. A continuity inequality for is derived and an iterative approximation for this ruin-related quantity is proposed. The results are illustrated by numerical examples.

Paper Structure

This paper contains 10 sections, 4 theorems, 65 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Introduction
  3. Definitions and Preliminaries
  4. Continuity estimate for ruin probability and for deficit at ruin in the classical risk model
  5. Ruin probability $\psi(u)$
  6. The deficit at ruin
  7. Ruin probability in a perturbed risk model
  8. Continuity inequality for $K(u)$
  9. Estimation for K(u)
  10. Conclusions

Key Result

Theorem 1

Let $X$ and $\widetilde{X}$ be two r.v.'s representing the individual claim amounts in two surplus processes as in risk process with distribution functions $F$ and $\widetilde{F}$, respectively. We assume that the intensities of claim arrivals are $\lambda$ and $\lambda_X$ and that $EX^{(\gamma+1)}< where,

Theorems & Definitions (11)

  • Theorem 1
  • Remark 1
  • Example 1
  • Theorem 2
  • Example 2
  • Remark 2
  • Theorem 3
  • Example 3
  • Lemma 1
  • Remark 3
  • ...and 1 more