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This work studies distributionally robust evaluation of expected functionvalues over temporal data. A set of alternative measures is characterized bythe causal optimal transport. We prove the strong duality and recast thecausality constraint as minimization over an infinite-dimensional test functionspace. We approximate test functions by neural networks and prove the samplecomplexity with Rademacher complexity. Moreover, when structural information isavailable to further restrict the ambiguity set, we prove the dual formulationand provide efficient optimization methods. Empirical analysis of realizedvolatility and stock indices demonstrates that our framework offers anattractive alternative to the classic optimal transport formulation.

Distributionally robust risk evaluation with a causality constraint and structural information