Let $M=(M_t)_{1\le t\le 2}$ be a continuous (resp. right-continuous) martingale. Denote $x:=\mathbb E[M_1]\in\mathbb R$. Can we construct on some probability space a continuous (resp. right-continuous) martingale $N=(N_t)_{0\le t\le 1}$ such that

$$N_0=x \quad \mbox{and} \quad \mbox{Law}(M_t: 1\le t\le 2)=\mbox{Law}(N_t: 1\le t\le 2)?$$

Let $M=(M_t)_{1\le t\le 2}$ be a continuous (resp. right-continuous) martingale. Denote $x:=\mathbb E[M_1]\in\mathbb R$. Can we construct on some probability space a continuous (resp. right-continuous) martingale $N=(N_t)_{0\le t\le 2}$ such that

$$N_0=x \quad \mbox{and} \quad \mbox{Law}(M_t: 1\le t\le 2)=\mbox{Law}(N_t: 1\le t\le 2)?$$