绝对破产情形下经典风险模型的最优分红和注资问题

Optimal Dividend and Capital Injection of the Classical Risk Model under Absolute Ruin

本文在绝对破产情形下考虑了经典风险模型的最优分红与注资问题。一方面当保险公司产生赤字时可向银行进行贷款,获得的保费将用来偿还银行,一旦盈余过程低于绝对破产线则破产发生。另一方面公司可采用存在比例交易费的注资以使股东权益最大化。本文的目的是最大化预期累积贴现分红与预期贴现注资成本之差的期望。我们给出了该模型下值函数V(x)的定义并对其进行刻画,证明了值函数 满足的基本性质。然后通过动态规划原理建立V(x)满足的HJB方程,进而证明为该HJB方程的几乎处处解,并证明了验证定理。

In this paper, we consider the optimal dividend and capital injection of the classical risk model un-der Absolute Ruin. On the one hand, when the insurance company has a deficit, it will make loans to the bank, and the premiums obtained will be used to repay the bank loans. Once the surplus pro-cess is below the absolute ruin line, bankruptcy will occur. On the other hand, the company can use the capital injection with proportional transaction fee for maximizing shareholders’ equity. The purpose of this paper is to maximize the expected difference between the expected cumulative dis-count dividend and the expected discounted capital injection cost. We give the definition of the val-ue function V(x) under this model and describe it and prove the basic properties that the value function V(x) satisfies. Then, we establish the satisfying HJB equation through the principle of dynamic programming and prove the solution of the HJB equation almost everywhere, and prove the verification theorem.