Extended Convolution Bounds on the Fr\'{e}chet Problem: Robust Risk Aggregation and Risk Sharing

In this paper, we provide extended convolution bounds for the Fr\'{e}chet problem and discuss related implications in quantitative risk management. First, we establish a new form of inequality for the Range-Value-at-Risk (RVaR). Based on this inequality, we obtain bounds for robust risk aggregation with dependence uncertainty for (i) RVaR, (ii) inter-RVaR difference and (iii) inter-quantile difference, and provide sharpness conditions. These bounds are called extended convolution bounds, which not only complement the results in the literature (convolution bounds in Blanchet et al. (2025)) but also offer results for some variability measures. Next, applying the above inequality, we study the risk sharing for the averaged quantiles (corresponding to risk sharing for distortion risk measures with special inverse S-shaped distortion functions), which is a non-convex optimization problem. We obtain the expression of the minimal value of the risk sharing and the explicit expression for the corresponding optimal allocation, which is comonotonic risk sharing for large losses and counter-comonotonic risk sharing for small losses or large gains. Finally, we explore the dependence structure for the optimal allocations, showing that the optimal allocation does not exist if the risk is not bounded from above.