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

## TL;DR

This paper investigates a single-period optimal transport problem with a backward martingale constraint and a covariance-type cost, providing conditions for optimality, uniqueness, and representation, inspired by insider trading models.

## Contribution

It introduces a novel optimal transport framework with backward martingale constraints and characterizes optimal plans via maximal monotone sets, extending classical models.

## Key findings

- Optimal plans are supported on maximal monotone sets.
- Sharp regularity conditions for uniqueness and map representation.
- Connection to insider trading models like the Kyle model.

## Abstract

We study a single-period optimal transport problem on $\mathbb{R}^2$ with a covariance-type cost function $c(x,y) = (x_1-y_1)(x_2-y_2)$ and a backward martingale constraint. We show that a transport plan $\gamma$ is optimal if and only if there is a maximal monotone set $G$ that supports the $x$-marginal of $\gamma$ and such that $c(x,y) = \min_{z\in G}c(z,y)$ for every $(x,y)$ in the support of $\gamma$. 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).