An optimal transport problem with backward martingale constraints motivated by insider trading

Abstract

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 γ is optimal if and only if there is a maximal monotone set G that supports the x-marginal of γ and such that c(x,y)=minz∈Gc(z,y) for every (x,y)∈suppγ. 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 (Rev. Econ. Stud. 61 (1994) 131–152).