This paper studies the optimal reinsurance‐investment problems for an insurance company where the claim process follows a Brownian motion with drift. It turns out that there is a region where the probability of drawdown, namely, the probability that the value of the insurer's surplus process reaches some fixed fractional value of its maximum value to date is positive. Then in the complementary region, drawdown can be avoided with certainty. For this reason, we call the former region the “danger‐zone” and “safe‐region” for the latter. In the danger‐zone, we consider the problem of minimizing the probability of drawdown; and in the safe‐region, we turn our attention to the optimization problem of minimizing the expected time to reach a given capital level. Using the technique of stochastic control theory and the corresponding Hamilton‐Jacobi‐Bellman equation, explicit expressions of the optimal reinsurance‐investment strategies and the associated value functions are derived for the two optimization problems. Moreover, we provide several detailed comparisons to investigate the impact of some important parameters on the optimal strategies and illustrate the observation from behavior finance of view.

中文翻译：

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在危险区域和安全区域的最佳再保险和投资

本文研究了保险公司的最优再保险投资问题，其中索赔过程遵循带漂移的布朗运动。事实证明，存在一个区域，其提款的概率（即，保险人的剩余过程的价值达到其最大值的某个固定的分数）的概率为正。然后，在互补区域中，可以确定地避免缩水。因此，我们将前一个区域称为“危险区域”，而将其称为“安全区域”。在危险区域中，我们考虑了使回撤概率最小化的问题。在安全区域，我们将注意力转向最优化问题，即缩短达到给定资本水平的预期时间。使用随机控制理论的技术和相应的Hamilton-Jacobi-Bellman方程，针对两个优化问题得出了最优再保险投资策略的显式表达式以及相关的价值函数。此外，我们提供了一些详细的比较，以调查一些重要参数对最佳策略的影响，并从行为财务的角度说明观察。