A martingale extension/interpolation problem

Question

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: $$ \mathbb{P}(\hat{X}_{t_n}\in \cdot) =\mathbb{P}\left(X_{t_n}\in \cdot\right)\qquad \forall n=0,\dots,N ? $$

Answer 1

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