On the ruin problem in the renewal risk processes perturbed by diffusion

TL;DR

This paper analyzes the ruin probabilities in perturbed renewal risk processes using integro-differential equations and Laplace transforms, providing explicit formulas and numerical illustrations for the Gerber-Shiu functions.

Contribution

It introduces explicit Laplace transform formulas for Gerber-Shiu functions in perturbed renewal risk models with claims from rational distributions.

Findings

Explicit Laplace transforms of Gerber-Shiu functions derived

Closed-form expressions obtained for rational claim distributions

Numerical examples illustrate theoretical results

Abstract

In this paper, we consider the perturbed renewal risk process. Systems of integro-differential equations for the Gerber-Shiu functions at ruin caused by a claim and oscillation are established, respectively. The explicit Laplase transforms of Gerber-Shiu functions are obtained, while the closed form expressions for the Gerber-Shiu functions are derived when the claim amount distribution is from the rational family. Finally, we present numerical examples intended to illustrate the main results.