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We investigate the problem of finding upper and lower bounds for a Choquetrisk measure of a nonlinear function of two risk factors, when the marginaldistributions of the risk factors are ambiguous and represented by nonadditivemeasures on the marginal spaces and the joint nonadditive distribution on theproduct space is unknown. We treat this problem as a generalization of theoptimal transport problem to the setting of nonadditive measures. We provideexplicit characterizations of the optimal solutions for finite marginal spaces,and we investigate some of their properties. We further discuss the connectionswith linear programming, showing that the optimal transport problems forcapacities are linear programs, and we also characterize their dualsexplicitly. Finally, we investigate a series of numerical examples, including acomparison with the classical optimal transport problem, and applications tocounterparty credit risk.

Bounds on Choquet Risk Measures in Finite Product Spaces with Ambiguous Marginals