An Optimization Dichotomy for Capital Injections and Absolutely Continuous Dividend Strategies

We study an optimal stochastic control problem in which a firm’s cash/surplus process is controlled by dividend payments and capital injections. We consider absolutely continuous dividend policies subject to a level-dependent upper bound on the dividend rate and general capital injection strategies. We construct an optimal solution for which either the optimal capital injections consist of a forced bailout strategy when the cash process reaches zero or no injection of capital is ever made and ruin is eventually reached. This gives rise to two distinct dividend optimization problems for which the solutions are shown to be mean-reverting dividend strategies refracted at optimal thresholds. To prove the existence of the optimal threshold in the forced-injection case, we use the theory of viscosity solutions and characterize the optimal threshold in terms of the derivative of the value function. By a uniqueness result for the solution of the associated HJB equation, we show that the value function corresponds to the performance function of a mean-reverting dividend strategy, which we compute explicitly using results from fluctuation theory, and we characterize the optimal threshold. Finally, we give a complete solution to the general problem and characterize the dichotomy by proving a comparison theorem based on the value functions (or their derivatives) at zero for the two dividend optimization problems. Funding: Funding in support of this work was provided by three Discovery Grants from the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2019-06538, RGPIN-2020-06619, RGPIN-2021-03299].