  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 which 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
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 non-overlapping 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.
