We study risk sharing games with quantile-based risk measures and heterogeneous beliefs, motivated by the use of internal models in finance and insurance. Explicit forms of Pareto-optimal allocations and competitive equilibria are obtained by solving various optimization problems. For Expected...

The Expected Shortfall (ES) is one of the most important regulatory risk measures in finance, insurance, and statistics, which has recently been characterized via sets of axioms from perspectives of portfolio risk management and statistics. Meanwhile, there is large literature on insurance...

The class of distortion riskmetrics is defined through signed Choquet integrals, and it includes many classic risk measures, deviation measures, and other functionals in the literature of finance and actuarial science. We obtain characterization, finiteness, convexity, and continuity results on...

In the recent Basel Accords, the Expected Shortfall (ES) replaces the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most important risk measure in financial regulation. One of the most challenging tasks in risk modeling practice is to...

Optimization of distortion riskmetrics with distributional uncertainty has wide applications in finance and operations research. Distortion riskmetrics include many commonly applied risk measures and deviation measures, which are not necessarily monotone or convex. One of our central findings is...