# Pointwise second moment of continuous local martingale

Let $$(M_t)_{t \ge 0}$$ be a continuous local martingale (as defined in LeGall). Let $$\mathbb{E}$$ denote expectation.

Is $$\mathbb{E}[M_t^2] < \infty$$ for all $$t \ge 0$$? If yes, then how does one prove it? If no, then how does one construct a counterexample?

My attempt: Fix $$\tau > 0$$. The sequence of stopping times $$T_n = n$$ reduces $$M$$ (from LeGall). Define a constant stopping time $$T=\tau$$. By the same argument, $$M^T$$, which is the martingale $$M_{t \wedge T}$$, is uniformly integrable, and therefore, bounded in $$L^1$$, which gives that $$M_{\tau} \in L^1$$.

Perhaps, I might be missing something very basic here, but I do not see the $$L^2$$ conclusion I am seeking appear naturally from here. Any hints or suggestions are appreciated.

$$M_t=B_tX$$ where $$X$$ is an integrable but not square-integrable random variable, and $$(B_t)_{t\ge0}$$ is a brownian motion independent of $$X$$.
If you allow $$M_t$$ not to start from $$0$$ then it is even more simple: $$M_t=X$$.