Confidence intervals of ruin probability under L\'evy surplus

TL;DR

This paper develops a method to construct confidence intervals for the ruin probability in insurance models driven by Le9vy processes, using parametric estimation and asymptotic approximations.

Contribution

It introduces a Crame9r-type approximation for the derivative of ruin probability, enabling explicit asymptotic confidence intervals under Le9vy surplus models.

Findings

Provides a practical method for confidence interval construction.

Addresses the challenge of derivative in asymptotic variance.

Offers theoretical justification for the approximation.

Abstract

The aim of this paper is to construct the confidence interval of the ultimate ruin probability under the insurance surplus driven by a L\'evy process. Assuming a parametric family for the L\'evy measures, we estimate the parameter from the surplus data and estimate the ruin probability via the delta method. However the asymptotic variance includes the derivative of the ruin probability with respect to the parameter, which is not generally given explicitly, and the confidence interval is not straightforward even if the ruin probability is well estimated. This paper gives the Cram\'er-type approximation for the derivative and gives an asymptotic confidence interval of ruin probability.