Backward martingale transport maps and equilibrium with insider

We consider an optimal transport problem with backward martingale constraint. The objective function is given by the scalar product of a pseudo-Euclidean space $S$. We show that the supremums over maps and plans coincide, provided that 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 a difference of convex functions) with strictly positive normal vectors in the sense 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 an application, we derive sharp conditions for the existence and uniqueness of equilibrium in a multi-asset version of the model with insider from Rochet and Vila [10]. In the linear-Gaussian case, we characterize Kyle’s lambda, the sensitivity of price to trading volume, as the unique positive solution of a non-symmetric algebraic Riccati equation.