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.