A FREE BOUNDARY PROBLEM OF LIQUIDITY MANAGEMENT FOR OPTIMAL DIVIDEND AND INSURANCE IN FINITE HORIZON

In this paper, we devote our attention to the liquidity and risk management of a firm who faces two types of risks: a Poisson risk that can be insured for a fair premium and a Brownian risk that cannot be hedged nor insured. We apply a PDE method to solve this problem; the associated HJB equation is a Barenblatt equation with a gradient constraint in finite horizon. We not only show the existence of a classical solution to the problem, but also characterize the properties of the free boundaries arising from the HJB equation; especially, we are able to give a sufficient and necessary condition for the existence of the insurance free boundary.