Dynamic Optimal Reinsurance and Dividend Payout in Finite Time Horizon

This paper studies a dynamic optimal reinsurance and dividend-payout problem for an insurance company in a finite time horizon. The goal of the company is to maximize the expected cumulative discounted dividend payouts until bankruptcy or maturity, whichever comes earlier. The company is allowed to buy reinsurance contracts dynamically over the whole time horizon to cede its risk exposure with other reinsurance companies. This is a mixed singular–classical stochastic control problem, and the corresponding Hamilton–Jacobi–Bellman equation is a variational inequality with a fully nonlinear operator and subject to a gradient constraint. We obtain the $C^{2,1}$ smoothness of the value function and a comparison principle for its gradient function by the penalty approximation method so that one can establish an efficient numerical scheme to compute the value function. We find that the surplus-time space can be divided into three nonoverlapping regions by a risk-magnitude and time-dependent reinsurance barrier and a time-dependent dividend-payout barrier. The insurance company should be exposed to a higher risk as its surplus increases, be exposed to the entire risk once its surplus upward crosses the reinsurance barrier, and pay out all its reserves exceeding the dividend-payout barrier. The estimated localities of these regions are also provided.

Funding: This work was supported by the Hong Kong Research Grants Council [Grants GRF 15202421 and GRF 15202817], the Guangdong Basic and Applied Basic Research Foundation [Grants 2021A1515012031 and 2022A1515010263], the National Natural Science Foundation of China [Grants 11901244 and 11971409], the PolyU-SDU Joint Research Center on Financial Mathematics, the CAS AMSS-PolyU Joint Laboratory of Applied Mathematics, and Hong Kong Polytechnic University.