Optimal Ratcheting of Dividends with Capital Injection

In this paper, we investigate the optimal dividend problem with capital injection and ratcheting constraint with nondecreasing dividend payout rate. Capital injections are introduced in order to eliminate the possibility of bankruptcy. Under the Cramér–Lundberg risk model, the problem is formulated as a two-dimensional stochastic control problem. By applying the viscosity theory, we show that the value function is the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation. In order to obtain analytical results, we further study the problem with finite ratcheting constraint, where the dividend rate takes only a finite number of available values. We show that the value function under general ratcheting can be approximated arbitrarily closely by the one with finite ratcheting. Finally, we derive the expressions of value function when the threshold-type finite ratcheting dividend strategy with capital injection is applied, and we show the optimality of such a strategy under certain conditions of concavity. Numerical examples under various scenarios are provided at the end.