On an insurance ruin model with a causal dependence structure and perturbation

The classical compound Poisson risk model and the Sparre-Andersen risk model for insurance businesses assume that the interclaim times and the claim amounts are independently distributed. To relax the independence assumption, we consider a continuous-time risk model under which the interclaim-time distribution depends on the size of the previously occurred claim. For practical purposes, the surplus process is further assumed to be perturbed by a Brownian motion to address small financial fluctuations or investment returns. To analyze the risk associated with the event when the insurer's ruin occurs, explicit solutions for the Gerber–Shiu discounted penalty function may be derived through the defective renewal equations provided here when claim amounts follow an arbitrary distribution. Applications with Kn family claim amounts and the Laplace transform of the ruin time are discussed in detail. An illustrative numerical example is presented to assess the impact of different perturbations on the underlying dependent-structure surplus process.