Inf-Convolution, Optimal Allocations, and Model Uncertainty for Tail Risk Measures
研究了尾部风险度量的风险分担优化问题，推导了无穷卷积和帕累托最优分配的显式公式，并在椭圆模型和模型不确定性下得到最优分配，结果不要求尾部风险度量具有凸性。
Inspired by the recent developments in risk sharing problems for the value at risk (VaR), the expected shortfall (ES), and the range value at risk (RVaR), we study the optimization of risk sharing for general tail risk measures. Explicit formulas of the inf-convolution and Pareto-optimal allocations are obtained in the case of a mixed collection of left and right VaRs, and in that of a VaR and another tail risk measure. The inf-convolution of tail risk measures is shown to be a tail risk measure with an aggregated tail parameter, a phenomenon very similar to the cases of VaR, ES, and RVaR. Optimal allocations are obtained in the settings of elliptical models and model uncertainty. In particular, several results are established for tail risk measures in the presence of model uncertainty, which may be of independent interest outside the framework of risk sharing. The technical conclusions are quite general without assuming any form of convexity of the tail risk measures. Our analysis generalizes in several directions the recent literature on quantile-based risk sharing.