Construct a martingale with two given distributions. This is a follow up of another post:
Construct a martingale with a given distribution?
Given two distributions $f_1(\cdot)$ and $f_2(\cdot)$ on $\mathbb R$, under what condition can we construct a martingale $X_t$, such that $X_1$ has distribution $f_1$, and $X_2$ has distribution $f_2$?
From martingale property, we know that $E(X_1)=E(X_2)$ and $Var(X_1)\le Var(X_2)$, so we know $f_1(\cdot)$ and $f_2(\cdot)$ can not be arbitrary. 
 A: The martingale property means that $\Bbb E[X_2|X_1]=X_1$. Jensen's inequality now implies that if $\varphi:\Bbb R\to\Bbb R$ is a convex function then
$$
\Bbb E[\varphi(X_1)]=\Bbb E[\varphi(\Bbb E[X_2|X_1])]\le \Bbb E[\Bbb E[\varphi(X_2)|X_1]]=\Bbb E[\varphi(X_2)].
$$
This "monotonicity along convex functions" property is not only necessary but also sufficient for two distributions to be the marginals at two times of a martingale. This result is due to V. Strassen, in The existence of probability measures with given marginals, at http://projecteuclid.org/download/pdf_1/euclid.aoms/1177700153 . Thus, you need to check that $\int_{\Bbb R}\varphi(x)f_1(x)\,dx\le\int_{\Bbb R}\varphi(x)f_2(x)\,dx $ for all convex $\varphi$.
A: Modulo technical details (on which I am no expert) the answer is Yes. The result is due to J.L. Doob ["Generalized sweeping-out and probability", J. Functional Analysis vol. 2, (1968) 207–225], who showed the analogous result for submartingales.   Strassen's work was applied to the problem by H. G. Kellerer "Markov-Komposition und eine Anwendung auf Martingale" dating to 1972, in Mathematische Annalen.  The topic is of some current research interest, under the name "peacock" (for the french acronym PCOC = Processus Croissants pour l'Ordre Convexe).
