Is a càdlàg martingale on a finite time horizon square-integrable?

Question

We work on a filtered probability space with finite time horizon $T$. Let $X$ be a càdlàg martingale.

Question: Is $X$ square-integrable, i.e. does $\sup_{t\in[0,T]}E[X_t^2] <\infty$ hold?

We know that $X_T^* := \sup_{t\in[0,T]}|X_t| <\infty$ a.s., but I am not sure if this helps.

Answer 1

No. Let $X \in L^1 \setminus L^2$ and set $X_t = X$ for $t \in [0,T]$. Then $\{X_t\}$ is continuous, but not square-integrable.