Analysis Of Viscosity Solution Of A Class Of HJB Equation And Its Application In The Insurance Problem

We consider an optimal impulse control problem on reinsurance, dividend and rein-vestment of an insurance company. To close reality, we add fixed and proportional trans-action costs to this problem. The value of the company is associated with the expectedpresent value of the net dividends pay out minus the net reinvestment capitals until theruin time. We concentrate on non-cheap proportional reinsurance. We prove the valuefunction is the viscosity solution of HJB equation corresponding to the control problem,and establish the regularity property of the viscosity solution under a weak assumption.We give a rigorous proof on the solvability of non-uniformly elliptic Dirichlet problemassociated it. Finally, we derive the value function and the optimal strategy of the controlproblem.