# Existence of martingales given some constraint on laws

Let $X=(X)_{0\le t\le 1}$ be a continuous martingale starting at $0$, then denote by $\mu$ and $\nu$ the probability laws of $\int_0^1X_t \mathrm{d}t$ and $X_1$. Then it is easy to see that the couple $(\mu,\nu)$ is increasing in convex order, i.e.

holds for all convex functions $f:\mathbb{R}\to \mathbb{R}$ of linear growth, see also “Peacocks and Associated Martingales, with Explicit Constructions” for further details. Now my question is given by probability measures $\mu$ and $\nu$ on $\mathbb{R}$, could we give some conditions on the couple $(\mu,\nu)$ such that there exists a continuous martingale $X=(X)_{0\le t\le 1}$ satisfying