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

Given a probability measure ν on R2, we want to minimize
c(x, y)dγ over γ ∈ Γ(ν)
for the covariance-type cost function c(x, y) = (y1 −x1)(y2 −x2), where Γ(ν) is the family of probability measure γ on R2 × R2, that have ν as their y-marginal and make a martingale from the canonical two-dimensional process (x, y). Problem (1) belongs to the class of optimal transport problems with backward martingale constraints, in the sense that the initial x-marginal is part of the solution. The motivation comes from a version of Kyle’s equilibrium with insider.
Our main result states that a probability measure γ ∈ Γ(ν) is optimal if and only if there is a maximal monotone set G ⊂ R2 such that (1) it supports the x-marginal of γ, and (2) c(x, y) = minz∈G c(z, y) for every (x, y) ∈ supp γ. Furthermore, if ν is continuous, then the solution is uniquely determined by the subdifferential of the concave function uG(y) = infz∈G{c(z, y) − y1y2}.

The presentation is based on a joint paper with Yan Xu available on
Paper: https://arxiv.org/abs/1906.03309.