A Laplace-based perspective on conditional mean risk sharing

TL;DR

This paper introduces a Laplace transform-based framework for efficiently computing conditional mean risk-sharing allocations in multivariate models, enabling closed-form solutions and improved numerical methods for complex distributions.

Contribution

It develops a novel approach using joint Laplace--Stieltjes transforms to compute CMRS allocations, including analytic and numerical inversion techniques, applicable to high-dimensional risk models.

Findings

Provides closed-form solutions for certain distributions

Develops numerical inversion methods for complex cases

Demonstrates effectiveness in high-dimensional settings

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 $\nu_i(B)=\mathbb{E}[X_i\boldsymbol{1}_{\{S\in B\}}]$ 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 $\xi_i$, and hence $h_i(s)=\xi_i(s)/f_S(s)$ 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.