Gerber–Shiu Penalty Function for a Dependent Risk Model with Stochastic Premium Income

Abstract

This study considers a dependent insurance risk model in which premium income is represented by a stochastic compound Poisson process and the claim mechanism incorporates dependence between claim amounts and subsequent inter-claim times through a stochastic threshold structure. The Gerber–Shiu expected discounted penalty function is investigated for two states determined by the relationship between the original loss amount and the corresponding stochastic threshold. Under the assumption that individual premium amounts follow an exponential distribution, integral equations for the state-dependent Gerber–Shiu functions are established and transformed using Laplace-transform techniques. The resulting expressions are analysed through a characteristic equation, from which the required transform quantities are determined. A defective renewal-equation representation of the Gerber–Shiu functions is subsequently derived using the transformed expressions and appropriate operator identities. Two numerical examples are considered to illustrate the analytical results. In the first example, both the premium and original loss amounts follow exponential distributions. In the second example, a mixture of exponential distributions is introduced for one component of the loss-threshold structure while the premium amount remains exponentially distributed. The resulting ruin probabilities decrease as the initial surplus increases, while differences between the two dependence states are also observed. The analysis provides a mathematical framework for examining ruin-related quantities in a dependent risk model where premium income is stochastic rather than deterministic.