On a Perturbed Risk Model with Time-Dependent Claim Sizes

We consider a risk model perturbed by a Brownian motion, where the individual claim sizes are dependent on the inter-claim times. We study the Gerber–Shiu functions when ruin is due to a claim or the jump-diffusion process. Integro-differential equations and Laplace transforms satisfied by the Gerber–Shiu functions are obtained. Then, it is shown that the expected discounted penalty functions satisfy defective renewal equations. Explicit expressions can be obtained for exponential claim sizes. Finally, a numerical example is provided to measure the impact of the various dependence parameters in the risk model on the ruin probabilities.