🤖 AI Summary
This study addresses the computational challenges in evaluating Conditional Mean Risk Sharing (CMRS) within continuous multivariate risk models, where high-dimensional integrals often hinder efficient implementation. The authors propose a novel framework that systematically employs the joint Laplace–Stieltjes transform to derive CMRS allocations. By taking partial derivatives along the diagonal direction, they obtain a transformed representation of the allocation measure, which is then inverted numerically using one-dimensional Laplace inversion enhanced with exponential tilting to stabilize tail-event estimation. This approach yields closed-form or semi-analytical solutions across a range of distributions—including high-dimensional settings—significantly improving both computational efficiency and numerical stability. Consequently, the method overcomes the feasibility limitations of conventional techniques in complex, high-dimensional scenarios.
📝 Abstract
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.