Number of claims and ruin time for a refracted risk process

TL;DR

This paper derives explicit formulas for the joint distribution of ruin time and claim count in a refracted risk model, enhancing understanding of risk processes with level-dependent behavior.

Contribution

It provides a novel explicit expression for the joint density of ruin time and claim number in a refracted risk process, using integro-differential equations and Lagrange's Expansion.

Findings

Explicit joint density formula derived

Method based on integro-differential equations

Applicable to risk models with refraction level

Abstract

In this paper, we consider a classical risk model refracted at given level. We give an explicit expression for the joint density of the ruin time and the cumulative number of claims counted up to ruin time. The proof is based on solving some integro-differential equations and employing the Lagrange's Expansion Theorem.