Backward and forward Wasserstein projections in stochastic order

Motivated by applications of stochastic orders in statistics and economics, we study metric projections onto cones in the Wasserstein space of probability measures, defined by stochastic orders. Duality theorems for backward and forward projections are established under mild conditions, which lay a foundation for investigating sampling of measures in stochastic orders. To demonstrate applications of these theorems beyond practice, particular attention is given to mathematical properties of convex order and subharmonic order. While backward and forward cones possess distinct geometric properties, strong connections between backward and forward projections can be obtained in the convex order case. Compared with convex order, the study of subharmonic order is subtler, particularly the existence and regularity of optimal mappings. In all cases, Brenier-Strassen type polar factorization theorems are proved, thus providing a full picture of the decomposition of optimal couplings between probability measures given by deterministic contractions (resp. expansions) and stochastic couplings. The factorization of convex order projection completes the decomposition obtained by Gozlan and Juillet, which builds a connection with Caffarelli's contraction theorem. A further noteworthy addition to earlier results is the decomposition in the subharmonic order case where the optimal mappings are characterized by volume distortion properties. To our knowledge, this is the first time in this occasion such results are available in the literature.