Optimal reinsurance and dividends with transaction costs and taxes under thinning structure

In this paper, we investigate the problem of optimal reinsurance and dividends under the Cramér–Lundberg risk model with the thinning-dependence structure which was first introduced by Wang and Yuen [Wang, G. & Yuen, K. C. (2005). On a correlated aggregate claims model with thinning-dependence structure. Insurance: Mathematics and Economics 36(3), 456–468]. The optimization criterion is to maximize the expected accumulated discounted dividends paid until ruin. To enhance the practical relevance of the optimal dividend and reinsurance problem, non-cheap reinsurance is considered and transaction costs and taxes are imposed on dividends. These realistic features convert our optimization problem into a mixed classical-impulse control problem. For the sake of mathematical tractability, we replace the Cramér–Lundberg risk model by its diffusion approximation. Using the method of quasi-variational inequalities, we show that the optimal reinsurance follows a two-dimensional excess-of-loss reinsurance strategy, and the optimal dividend strategy turns out to be an impulse dividend strategy with an upper and a lower barrier, i.e. everything above the lower barrier is paid as dividends whenever the surplus goes beyond the upper barrier, and no dividends are paid otherwise. Under the diffusion risk model, closed-form expressions for the value function associated with the optimal dividend and reinsurance strategy are derived. In addition, some numerical examples are presented to illustrate the optimality results.