SIAM Journal on Control and Optimization, Volume 59, Issue 4, Page 2524-2545, January 2021.
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.

中文翻译：

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有限视野下最优股息和保险的流动性管理的一个自由边界问题

SIAM Journal on Control and Optimization，第 59 卷，第 4 期，第 2524-2545 页，2021
年1 月。在本文中，我们专注于面临两种风险的公司的流动性和风险管理：泊松风险为公平的保费和无法对冲或保险的布朗风险投保。我们应用 PDE 方法来解决这个问题；关联的 HJB 方程是一个 Barenblatt 方程，在有限范围内具有梯度约束。我们不仅证明了该问题的经典解的存在，而且还刻画了由 HJB 方程产生的自由边界的性质；特别是，我们能够给出无保险边界存在的充分必要条件。