An Optimal Transport Problem with Backward Martingale Constraint Motivated by Equilibrium with Insider

We study a single-period optimal transport problem on R2 with a covariance-type cost function c(x; y) = (x1 - y1)(x2 - y2) and a backward martingale constraint. We show that a transport plan is optimal if and only if there is a maximal monotone set G that supports the x-marginal of and such that c(x; y) = minz2G c(z; y) for every (x; y) E supp y. We obtain sharp regularity conditions for the uniqueness of an optimal plan and for its representation in terms of a map. Our study is motivated by a variant of the classical Kyle model of insider trading from Rochet and Vila (1994).