基于拉普拉斯视角的条件均值风险分摊
A Laplace-based perspective on conditional mean risk sharing
AI总结：
本文提出基于拉普拉斯变换的条件均值风险分摊计算框架，通过联合拉普拉斯-斯蒂尔切斯变换实现高效分配，适用于高维和复杂分布场景。
摘要：
条件均值风险分摊（CMRS）规则是将总体损失分配到个体风险中的重要工具，但其在连续多变量模型中的实施通常需要复杂的多维积分。我们开发了一个框架，通过风险向量的联合拉普拉斯-斯蒂尔切斯变换计算CMRS分配。分配措施ν_i(B)=E[X_i1{S∈B}]的拉普拉斯变换可以表示为联合拉普拉斯变换在对角线t1=…=tn上的偏导数。当密度存在时，这会产生一维拉普拉斯反演用于f_S和ξ_i，从而得到h_i(s)=ξ_i(s)/f_S(s)在绝对连续部分上的表达式，为广泛分布类提供闭式或半解析解。我们还开发了数值反演方法，用于无法获得解析反演的情况。我们引入了指数倾斜程序以稳定低概率总体事件的数值反演。我们提供了几个例子来说明该方法，包括在某些高维设置中，现有方法不可行的情况。
英文摘要：
The conditional mean risk-sharing (CMRS) rule is an important tool for distributing aggregate losses across individual risks, but its implementation in continuous multivariate models typically requires complicated multidimensional integrals. We develop a framework to compute CMRS allocations from the joint Laplace--Stieltjes transform of the risk vector. The LSTs of the allocation measures $ν_i(B)=\mathbb{E}[X_i\boldsymbol{1}_{\{S\in B\}}]$ are expressed as partial derivatives of the joint LST evaluated on the diagonal $t_1=\cdots=t_n$. When densities exist, this yields one-dimensional Laplace inversions for $f_S$ and $ξ_i$, and hence $h_i(s)=ξ_i(s)/f_S(s)$ on the absolutely continuous part, providing closed-form or semi-analytic solutions for a broad class of distributions. We also develop numerical inversion methods for cases where analytic inversion is unavailable. We introduce an exponential tilting procedure to stabilize numerical inversion in low-probability aggregate events. We provide several examples to illustrate the approach, including in some high-dimensional settings where existing approaches are infeasible.
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