Consider a non-constant martingale $(M_n)_{n\ge 0}$ with the filtration $(\mathcal{F}_n)_{n\ge 0}$, where $\mathcal{F}_n=\sigma(M_m:m\le n)$.
Define for $t\in\mathbb{R}_{\ge 0}$:
$$X_t:=M_{\lfloor t\rfloor}+(t-\lfloor t\rfloor)(M_{\lceil t\rceil}-M_{\lfloor t\rfloor})$$
How does the natural filtration $(\mathcal{A}_t)_{t\ge 0}$ with $\mathcal{A}_t=\sigma(X_s:s\le t)$ look like? And is $X=(X_t)_{t\ge 0}$ a martingale with respect to $(\mathcal{A}_t)_{t\ge 0}$? Any help or hint will be appreciated.
Thanks in advance!