Optimal dividends with debts and nonlinear insurance risk processes

The optimal dividend problem is a classic problem in corporate finance though an early contribution to this problem can be traced back to the seminal work of an actuary, Bruno De Finetti, in the late 1950s. Nowadays, there is a leap of literature on the optimal dividend problem. However, most of the literature focus on linear insurance risk processes which fail to take into account some realistic features such as the nonlinear effect on the insurance risk processes. In this paper, we articulate this problem and consider an optimal dividend problem with nonlinear insurance risk processes attributed to internal competition factors. We also incorporate other important features such as the presence of debts, constraints in regular control variables, fixed transaction costs and proportional taxes. This poses some theoretical challenges as the problem becomes a nonlinear regular-impulse control problem. Under some suitable hypotheses for the value function, we obtain the structure of the value function using its properties, without guessing its structure, which is widely used in the literature. By solving the corresponding Hamilton–Jacobi–Bellman (HJB) equation, closed-form solutions to the problem are obtained in various cases.