Let $(\Omega,\mathcal{F},(\mathcal{F}\_t)\_{t\geq 0},\mathbb{P})$ be a stochastic basis and let $N\in\mathbb{Z}^+$, $T>0$, $\{t\_n\}\_{n=1}^{N}$ be a partition of $[0,T]$ with $t\_0=0,t\_n<t\_{n+1},t\_N=T$. Fix $k\in \mathbb{Z}^+$, and suppose that for each $n=0,\dots,N$ we are given some
$$
X\_{t\_n}\in L^1\_{\mathbb{P}}(\mathcal{F}\_{t\_n-},\mathbb{R}^k).
$$


Then, *(or when?)* can we find an $(\mathcal{F}\_t)\_{t\geq 0}$-adapted martingale $(\hat{X}\_t)\_{t\geq 0}$ with $\sup\_{t\in [0,T]} \mathbb{E}\_{\mathbb{P}}[\|\hat{X}\_t\|^2]<\infty$ and with:
$$
\mathbb{P}(\hat{X}\_{t\_n}\in \cdot) =\mathbb{P}\left(X\_{t\_n}\in \cdot\right)\qquad \forall n=0,\dots,N
?
$$



A martingale extension/interpolation problem
Assume the filtration is large enough. Then Kellerer and Strassen both proved indepently the existence of such a martingale is equivalent to


$$\int fd\mu\_{t\_n} \le \int fd\mu\_{t\_{n+1}},\quad \mbox{for all convex functions} f:\mathbb R^k \to\mathbb R \mbox{ with linear growth},~~~~~(\ast)$$


where $\mu\_{t\_n}$ denotes the law of $X\_{t\_k}$. Here the relation $(\ast)$ is called increasing convex order. In particular, $\sup\_{0\le t\le T}\mathbb E[\|\hat X\_{t}\|^2]<\infty$ is satisfied iff $\mu\_{T}$ admits a finite second moment.


See e.g.


<https://www.esaim-ps.org/articles/ps/pdf/2012/01/ps110065.pdf>
<https://www.maths.univ-evry.fr/prepubli/361.pdf>
<https://hal.archives-ouvertes.fr/hal-02123427/document>


