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We consider an optimal transport problem with backward martingale constraint.The objective function is given by the scalar product of a pseudo-Euclideanspace $S$. We show that the supremums over maps and plans coincide, providedthat the law $\nu$ of the input random variable $Y$ is atomless. An optimal map$X$ exists if $\nu$ does not charge any $c-c$ surface (the graph of adifference of convex functions) with strictly positive normal vectors in thesense of the $S$-space. The optimal map $X$ is unique if $\nu$ does not charge$c-c$ surfaces with nonnegative normal vectors in the $S$-space. As anapplication, we derive sharp conditions for the existence and uniqueness ofequilibrium in a multi-asset version of the model with insider from Rochet andVila [10]. In the linear-Gaussian case, we characterize Kyle's lambda, thesensitivity of price to trading volume, as the unique positive solution of anon-symmetric algebraic Riccati equation.

Backward martingale transport maps and equilibrium with insider