Optimal Ratcheting of Dividends with Irreversible Reinsurance

This paper considers an insurance company that faces two key constraints: a ratcheting dividend constraint and an irreversible reinsurance constraint. The company allocates part of its reserve to pay dividends to its shareholders while strategically purchasing reinsurance for its claims. The ratcheting dividend constraint ensures that dividend cuts are prohibited at any time. The irreversible reinsurance constraint ensures that reinsurance contracts cannot be prematurely terminated or sold to external entities. The dividend rate level and the reinsurance level are modelled as nondecreasing processes, thereby satisfying the constraints. The incurred claims are modelled via a Brownian risk model. The main objective is to maximize the cumulative expected discounted dividend payouts until the time of ruin. The reinsurance and dividend levels belong to either a finite set or a closed interval. The optimal value functions for the finite set case and the closed interval case are proved to be the unique viscosity solutions of the corresponding Hamilton-Jacobi-Bellman equations, and the convergence between these optimal value functions is established. For the finite set case, a threshold strategy is proved to be optimal, while for the closed interval case, an $\epsilon$-optimal strategy is constructed. Finally, numerical examples are presented to illustrate the optimality conditions and optimal strategies.