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 and reinsurance level are modeled as nondecreasing processes, thereby satisfying the constraints. Claims are modeled using 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 are restricted to a finite set. The optimal value function is shown to be the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. A threshold strategy is constructed and shown to be optimal. Finally, numerical examples are presented to illustrate the optimality conditions and optimal strategies.