An injective martingale coupling

TL;DR

The paper addresses constructing a strongly injective martingale coupling for any pair of measures $μ,ν$ with $μ leq_{cx} ν$ and $ν$ continuous, ensuring that each target value $y$ is sourced from a unique initial point $x$ in the disintegration. The authors fuse shadow/left-curtain machinery with a careful decomposition into irreducible components and countably many intervals, yielding an explicit, constructive procedure. They develop a robust building block in the regular, atomless setting and extend it to the general continuous case via a sequence of interval-wise injections and patching, establishing the existence of strongly injective couplings and, in the continuous-by-continuous special case, even stronger structural properties (e.g., $ ext{supp}_I( obracket π_x)= ext{supp}( obracket π_x)$ for all $x$). The results enrich the martingale transport literature by providing a concrete, modular construction of injective couplings, clarifying the interplay between convex order, shadow measures, and curtain-couplings, and offering tools potentially relevant for martingale optimal transport and related theories.

Abstract

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

Figures (6)

• Figure 1: Sketch of the densities $\rho_\mu$ and $\rho_\nu$ and the locations of $f = f (x)$, $h = h(x)$ for given $x\in(x_0=x_*,x_1)$. Mass in $(x_0= x_*,x)$ according to the initial law in is mapped to the interval $(f(x),h(x))$ according to the target law. In particular, at the margins, mass at $x$ is mapped to $f(x)$ and $h(x)$ in a way which respects the martingale property.
• Figure 2: Stylized plots of functions $f$ and $h$ (on $[x_2,x_3]$) that support the injective coupling of Example \ref{['eg:mixeduniform']}. Note that $h$ (resp. $f$) is non-decreasing (resp. non-increasing) on $[x_0,x_1]$, non-increasing (resp. non-decreasing) on $[x_2,x_0]$ and again non-decreasing (resp. non-increasing) on $[x_1,x_3]$.
• Figure 3: The graphs of a continuous $y\mapsto\mathcal{H}(y)$ and its convex hull $\mathcal{H}^c$. The dashed (resp. dotted) curve represents $\mathcal{H}$ on $\{\mathcal{H}>\mathcal{H}^c\}$ (resp. on $\{\mathcal{H}=\mathcal{H}^c\}$), while the solid lines correspond to $\mathcal{H}^c$ on disjoint intervals that belong to $\{\mathcal{H}>\mathcal{H}^c\}$. In the figure, $z\in\mathbb R$ is such that $\mathcal{H}(z)>\mathcal{H}^c(z)$, and then $\mathcal{H}^c$ is linear on $(X^-_{\mathcal{H}}(z),Z^-_{\mathcal{H}}(z)=Z^+_{\mathcal{H}}(z))\supset(X^+_{\mathcal{H}}(z),Z^-_{\mathcal{H}}(z)=Z^+_{\mathcal{H}}(z))\ni z$ with slope $(\mathcal{H}^c)'(z)=\psi_+(z)=\psi_-(z)$.
• Figure 4: The construction of $\overrightarrow{m}_{x_1,x_2}$ and $\overrightarrow{n}_{x_1,x_2}$. For $x_1<x_2<u_1<u_2$, the dotted curve represents $\mathcal{E}_{x_1,u_2}$, the dashed curve corresponds to $\mathcal{E}_{x_1,u_1}$, while the dash-dotted curve represents $\mathcal{E}_{x_1,x_2}$. The solid curve corresponds to $D_{\mu,\nu}$. Note that $D_{\mu,\nu}\leq\mathcal{E}_{x_1,x_2}\leq\mathcal{E}_{x_1,u_1}\leq\mathcal{E}_{x_1,u_2}$ everywhere, $\mathcal{E}_{x_1,w}=D_{\mu,\nu}$ on $[x_1,w]$ for $w\in\{x_2,u_1,u_2\}$, and $\mathcal{E}_{x_1,u_i}=\mathcal{E}_{x_1,x_2}$ on $(-\infty,x_1]$ for $i=1,2$. Furthermore, for $i=1,2$, the straight line going through $(\overrightarrow{m}_{x_1,x_2}(u_i),\mathcal{E}_{x_1,u_i}(\overrightarrow{m}_{x_1,x_2}(u_i)))$ and $(\overrightarrow{n}_{x_1,x_2}(u_i),\mathcal{E}_{x_1,u_i}(\overrightarrow{n}_{x_1,x_2}(u_i)))$ corresponds to the linear section on $[\overrightarrow{m}_{x_1,x_2}(u_i),\overrightarrow{n}_{x_1,x_2}(u_i)]$ of the convex hull $\mathcal{E}^c_{x_1,u_i}$ of $\mathcal{E}_{x_1,u_i}$. In particular, $\overrightarrow{m}_{x_1,x_2}(u_2)\leq\overrightarrow{m}_{x_1,x_2}(u_1)\leq\overrightarrow{n}_{x_1,x_2}(u_1)\leq\overrightarrow{n}_{x_1,x_2}(u_2)$.
• Figure 5: Plot of $D_{\mu,\nu}$ for $(\mu,\nu) \in \mathcal{K}_R$. The line $L_{x_0}$ satisfies $D_{\mu,\nu}>L_{x_0}$ on $(-\infty,\underline{x}_0)$, $D_{\mu,\nu}=L_{x_0}$ on $[\underline{x}_0,x_0]$ and $D_{\mu,\nu}<L_{x_0}$ on $(x_0,\infty)$. On the other hand, the line $L_b$, that is tangent to $D_{\mu,\nu}$ at $b<\underline{x}_0$, crosses $D_{\mu,\nu}$ at $c>x_0$ and satisfies $D_{\mu,\nu}\geq L_{b}$ on $(-\infty,c]$ and $D_{\mu,\nu}< L_{b}$ on $(c,\infty)$.

Theorems & Definitions (114)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Remark 1.4
• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Example 2.5
• Example 2.6