Asymptotic behavior of the Gerber–Shiu discounted penalty function in the Erlang(2) risk process with subexponential claims

Jelena Kočetova,Jonas Šiaulys

doi:10.15388/na.16.3.14095

Abstract

We investigate the asymptotic behavior of the Gerber–Shiu discounted penalty function ɸ(u) = E(e−δT 1{T <∞} | U(0) = u), where T denotes the time to ruin in the Erlang(2) risk process. We obtain an asymptotic expression for the discounted penalty function when claim sizes are subexponentially distributed.

Highlights

In this paper we consider the insurer’s surplus process {U (t), t the equality N (t)U (t) = u + ct − Yi, i=10} which is defined by (1)where u ≥ 0 is the insurer’s initial surplus, c > 0 is the rate of premium income per unit time, and {N (t)}t≥0 is the renewal counting process for the number of claims up to time t
We investigate the asymptotic behavior of the Gerber–Shiu discounted penalty function φ(u) = E e−δT 1{T
I=1 where θ1, θ2, . . . is a sequence of independent and identically distributed random variables, which represent the inter-arrival times, with θ1 being the time until the first claim

Summary

Introduction

Where u ≥ 0 is the insurer’s initial surplus, c > 0 is the rate of premium income per unit time, and {N (t)}t≥0 is the renewal counting process for the number of claims up to time t. Cheng and Tang [2] derived some asymptotic formulas for the moments of the surplus prior to ruin and deficit at ruin in the renewal risk model with convolution-equivalent claim sizes and Erlang(2) inter-arrival times. Tang and Wei [9] studied the renewal risk model with absolutely continuous claim sizes whose density function belongs to some class of heavy-tailed distributions and satisfies some additional conditions. They obtained asymptotic formulas for the Gerber–Shiu discounted penalty function which involve the ladder heights and related quantities in the main terms. We can check that the main formula of Theorem 2 coincides with the asymptotic formula (3.15) of Tang and Wei [9], which was proved for absolutely continuous claim distributions

Proof of Theorem 1

Proof of Theorem 2

Examples