Dividend maximization in a diffusion-perturbed classical risk process compounded by proportional and excess-of-loss reinsurance

We study an optimal dividend problem for an insurance company whose surplus is modelled by a diffusion-perturbed classical risk process. The company chooses to enter into reinsurance treaties involving a combination of proportional and excess-of-loss reinsurance arrangements, and is allowed to pay dividends to the shareholders. Our main objective is to find an optimal dividend and reinsurance policy that maximizes the total expected discounted dividend payouts. We derive the Hamilton-Jacobi-Bellman equation and transform the resulting Volterra integrodifferential equation into a Volterra integral equation of the second kind. This integral equation is then solved numerically using the block-by-block method to determine the dividend and reinsurance strategies that optimize the dividend payouts to the shareholders. Numerical examples with both light- and heavy-tailed distributions in the diffusion case are given. We have obtained the optimal dividend barriers that maximize the total expected discounted dividend payouts. For the diffusion-perturbed model, the results show that the optimal reinsurance policy is not to reinsure.