An injective martingale coupling

Abstract

We give an injective martingale coupling; in particular, given measures $\mu$ and $\nu$ in convex order on $\mathbb R$ such that $\nu$ is continuous, we construct a martingale transport such that for each $y$ in the support of the target law $\nu$ there is a {\em unique} $x$ in {a support of} the initial law $\mu$ such that (some of) the mass at $x$ is transported to $y$. Then $\pi$ has disintegration $\pi(dx,dy) = \nu(dy) \delta_{\theta(y)}(dx)$ for some function $\theta$. More precisely we construct a martingale coupling $\pi$ of the measures $\mu$ and $\nu$ such that there is a set $\Gamma_\mu$ such that $\mu(\Gamma_\mu)=1$ and a disintegration $(\pi_x)_{x \in \Gamma_\mu}$ of $\pi$ of the form $\pi(dx,dy) = \pi_x(dy) \mu(dx)$ such that, with $\Gamma_{\pi_x}$ a support of $\pi_x$, we have $\# \{ x \in \Gamma_\mu : y \in \Gamma_{\pi_x} \} \in \{ 0,1 \}$ for all $y$ and $\{ y : \# \{ x \in \Gamma_\mu : y \in \Gamma_{\pi_x} \} = 1 \} = supp(\nu)$. Moreover, if $\mu$ is continuous we may take $\Gamma_{\pi_x} = supp(\pi_x)$ for each $x$. However, we cannot also insist that $\Gamma_\mu = supp (\mu)$.