We employ scoring functions, used in statistics for eliciting risk functionals, as cost functions in the Monge-Kantorovich (MK) optimal transport problem. This gives raise to a rich variety of novel asymmetric MK divergences, which subsume the family of Bregman-Wasserstein divergences. We show that for distributions on the real line, the comonotonic coupling is optimal for the majority the new divergences. Specifically, we derive the optimal coupling of the MK divergences induced by functionals including the mean, generalised quantiles, expectiles, and shortfall measures. Furthermore, we show that while any elicitable law-invariant convex risk measure gives raise to infinitely many MK divergences, the comonotonic coupling is simultaneously optimal. The novel MK divergences, which can be efficiently calculated, open an array of applications in robust stochastic optimisation. We derive sharp bounds on distortion risk measures under a Bregman-Wasserstein divergence constraint, and solve for cost-efficient portfolio strategies under benchmark constraints.

翻译：我们采用统计中用于诱导风险泛函的评分函数，作为Monge-Kantorovich（MK）最优传输问题中的代价函数。这产生了一类丰富的新型非对称MK散度，其包含了Bregman-Wasserstein散度族。我们证明：对于实直线上的分布，共单调耦合对大多数新散度而言是最优的。具体而言，我们推导了由均值、广义分位数、期望分位数及短缺测度等泛函诱导的MK散度的最优耦合。此外，我们证明：虽然任何可诱导的律不变凸风险测度都能产生无穷多个MK散度，但共单调耦合同时是最优的。这些可高效计算的新型MK散度为鲁棒随机优化开辟了广泛的应用前景。我们推导了Bregman-Wasserstein散度约束下扭曲风险测度的严格上界，并求解了基准约束下的成本最优投资组合策略。