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

Dmitry Kramkov and Yan Xu

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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).

Date: 2019-06
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Published in Annals of Applied Probability 2022, Vol. 32, No. 1, 294-326

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