Maximizing expected terminal utility of an insurer with high gain tax by investment and reinsurance

This paper investigates optimal investment and proportional reinsurance policies for an insurer who subjects to pay high gain tax. The surplus process of the insurer and the return process of the financial market are both modulated by the external macroeconomic environment. The dynamic of the external macroeconomic environment is specified by a Markov chain with finite states. Once the insurer’s accumulated profits attain a new maximum, they have to pay high gain tax. The objective of the insurer is to maximize the expected terminal utility by investment and reinsurance. The controlled wealth process of the insurer turned out to be a controlled jump diffusion process with reflections and Markov regime switching. By the weak dynamic programming principle (WDPP), we prove that the value function is the unique viscosity solution to the coupled Hamilton-Jacob-Bellman (HJB) equations with first derivative boundary constraints. By the Markov chain approximating method for the HJB equations, we construct a numerical scheme for approximating the viscosity solution to the coupled HJB equations. Two numerical examples are presented to illustrate the impact of both high gain tax and regime switching on the optimal policies.