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. Funding W. Wang was supported by the National Natural Science Foundation of China [Grants 12171405, 12271066, and 11661074] and the Fundamental Research Funds for the Central Universities of China [Grant 20720220044]. R. Xu was supported by the National Natural Science Foundation of China [Grants 12201506 and 12371468], the Natural Science Foundation of the Jiangsu Higher Education Institutions of China [Grant 21KJB110024], and Xi’an Jiaotong-Liverpool University Research Development Funding [Grant RDF-20-01-02].