Abstract

In this paper, we investigate the optimal dividend problem with capital injection and ratcheting constraint on the dividend payout rate, that is, the dividend payment process is absolutely continuous with non-decreasing dividend rate through the lifetime of the company. Capital injections with proportional transaction costs 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 optimal control problem. By applying the viscosity theory, we show that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation within certain functional class. However, analytical results of the value function and corresponding optimal strategy is rather difficult to obtain for general claim size distribution and general ratcheting of dividend strategy. Hence, in turn, we study the problem with finite ratcheting, where the finite ratcheting refers to the case when dividend rate takes only finite number of available values. With a convergence analysis, we show that the value function under general ratcheting constraint can be approximated arbitrarily closely by the ones with finite ratcheting assumption. Finally, under exponential jump sizes and finite ratcheting constraint, we derive analytical expressions for the value function when the threshold-type dividend strategies with capital injection are applied, and show that the optimal finite ratcheting strategy can be identified within such type, followed by some numerical examples illustrating the optimal value functions and the associated optimal strategies in various scenarios.