  We investigate the problem of finding upper and lower bounds for a Choquet
risk measure of a nonlinear function of two risk factors, when the marginal
distributions of the risk factors are ambiguous and represented by nonadditive
measures on the marginal spaces and the joint nonadditive distribution on the
product space is unknown. We treat this problem as a generalization of the
optimal transport problem to the setting of nonadditive measures. We provide
explicit characterizations of the optimal solutions for finite marginal spaces,
and we investigate some of their properties. We further discuss the connections
with linear programming, showing that the optimal transport problems for
capacities are linear programs, and we also characterize their duals
explicitly. Finally, we investigate a series of numerical examples, including a
comparison with the classical optimal transport problem, and applications to
counterparty credit risk.
