The Gerber-Shiu Function for a Class of Dependent Risk Models Perturbed by Brownian Motion
Bohui Wang *
School of Mathematics, Liaoning Normal University, Dalian, Liaoning 116081, China.
*Author to whom correspondence should be addressed.
Abstract
This study investigates a class of dependent insurance risk models perturbed by Brownian motion and incorporating a random claim-payment threshold. The insurer’s actual indemnity is defined as the minimum of the claim size and the threshold, allowing the model to represent partial claim compensation. The dependence structure is determined by two claim classes according to whether the claim amount exceeds the threshold, with the corresponding inter-claim times following exponential distributions with class-specific parameters. The analysis focuses on the generalised Gerber–Shiu function and distinguishes ruin caused by a claim from ruin caused by oscillation of the diffusion-perturbed surplus process. For both ruin mechanisms, systems of integro-differential equations are derived and their Laplace-transform representations are obtained. A generalised Lundberg equation is then used to characterise the relevant non-negative roots required for the subsequent analysis. On this basis, defective renewal equations are established for the Gerber–Shiu functions associated with the two initial classes. The resulting formulation extends a Brownian-perturbed dependent risk framework by incorporating a claim-payment threshold and provides a unified analytical representation of ruin-related quantities under partial indemnity. The study therefore develops a theoretical basis for examining the effects of threshold-based claim compensation within a dependent risk model subject to diffusion perturbation.
Keywords: Gerber–Shiu function, dependent risk model, Brownian motion, claim-payment threshold, partial indemnity, ruin by claims, ruin by oscillation, integro-differential equations, Laplace transform, defective renewal equation