Let me reformulate my question. Let $(X_0,X_T)$ be a martingale on $\mathbb R$, then it is known that one has a SDE:
$$Z_t=Z_0+\int_0^t\sigma(s,Z_s)dB_s, \mbox{ for all } t\in [0,T]~~~~~~~~~~~~~~(\ast)$$
s.t. $Z_0\sim X_0$ and $Z_T\sim X_T$. My question is the following: Could we construct a similar SDE above s.t.
$$(Z_0,Z_T)~~\sim~~(X_0,X_T)?$$
PS: Some references: https://www.maths.univ-evry.fr/prepubli/334.pdf