巨灾保险模型中的最优单调均值-方差问题
Optimal Monotone Mean-Variance Problem in a Catastrophe Insurance Model
AI总结：
本文在巨灾保险模型下，利用单调均值-方差准则研究最优投资与再保险问题，通过零和博弈和HJBI方程获得显式最优策略与有效前沿。
摘要：
本文研究一个同时涉及普通保险业务和巨灾保险业务的最优投资与再保险问题。巨灾事件被建模为复合泊松过程，并对普通保险业务产生影响。普通保险业务的索赔强度采用具有散粒噪声强度的Cox过程来描述，其跳跃幅度与巨灾事件的规模成正比。该强度在巨灾发生时上升，随后随时间衰减。保险人的目标是在单调均值-方差（MMV）准则下最大化其终端财富。与经典的均值-方差（MV）准则相比，MMV准则在其整个定义域上具有单调性，更符合基本经济学原理。我们首先将原始MMV优化问题表述为一个辅助的零和博弈。通过求解Hamilton-Jacobi-Bellman-Isaacs（HJBI）方程，得到了值函数和最优策略的显式形式。此外，我们给出了MMV准则下的有效前沿。文中给出了若干数值例子以展示结果的实际意义。
英文摘要：
This paper explores an optimal investment and reinsurance problem involving both ordinary and catastrophe insurance businesses. The catastrophic events are modeled as following a compound Poisson process, impacting the ordinary insurance business. The claim intensity for the ordinary insurance business is described using a Cox process with a shot-noise intensity, the jump of which is proportional to the size of the catastrophe event. This intensity increases when a catastrophe occurs and then decays over time. The insurer's objective is to maximize their terminal wealth under the Monotone Mean-Variance (MMV) criterion. In contrast to the classical Mean-Variance (MV) criterion, the MMV criterion is monotonic across its entire domain, aligning better with fundamental economic principles. We first formulate the original MMV optimization problem as an auxiliary zero-sum game. Through solving the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation, explicit forms of the value function and optimal strategies are obtained. Additionally, we provides the efficient frontier within the MMV criterion. Several numerical examples are presented to demonstrate the practical implications of the results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
