Let $\mu$, $\nu$ and $\gamma$ be three probability measures on $\mathbb R$. Consider the optimisation problem as follows:

$$\inf_{(X,Y,Z)}~ \mathbb E\big[|Y-Z|^2\big],$$

where the inf is taken overall $(X,Y,Z)$ such that $X\sim\mu$, $Y\sim\nu$, $Z\sim\gamma$ and $\mathbb E[Y|X]=X$, $\mathbb E[Z|X]=X$ (Or namely, $(X,Y)$ and $(X,Z)$ are martingales). It follows by Strassen's theorem, the necessary and sufficient condition for the existence of such triplet $(X,Y,Z)$ is that $\mu\preceq \gamma$ and $\nu\preceq \gamma$, i.e.

$$\int f d\mu \le \int f d\nu \mbox{ and } \int f d\mu \le \int f d\gamma, \quad \mbox{ for all convex functions } f:\mathbb R\to\mathbb R.$$

I am very interested in the optimiser of this problem. Any answer or comment will be highly appreciated!

Let $\mu$, $\nu$ and $\gamma$ be three probability measures on $\mathbb R$. Consider the optimisation problem as follows:

$$\inf_{(X,Y,Z)}~ \mathbb E\big[|Y-Z|^2\big],$$

where the inf is taken overall $(X,Y,Z)$ such that $X\sim\mu$, $Y\sim\nu$, $Z\sim\gamma$ and $\mathbb E[Y|X]=X$, $\mathbb E[Z|X]=X$ (Or namely, $(X,Y)$ and $(X,Z)$ are martingales). It follows by Strassen's theorem, the necessary and sufficient condition for the existence of such triplet $(X,Y,Z)$ is that $\mu\preceq \gamma$ and $\mu\preceq \nu$, i.e.

$$\int f d\mu \le \int f d\nu \mbox{ and } \int f d\mu \le \int f d\gamma, \quad \mbox{ for all convex functions } f:\mathbb R\to\mathbb R.$$

I am very interested in the optimiser of this problem. Any answer or comment will be highly appreciated!