Consider a bounded, continuous martingale $(X_t)_{t\ge 0}$. I was able to show that $(X^2-[X])_{t\ge 0}$ is uniformly integrable, where $[X]$ denotes the quadratic variation.
Is there an example of a bounded, continuous martingale $(X_t)_{t\ge 0}$, such that $(X^2-[X])_{t\ge 0}$ is unbounded, but still uniformly integrable?
I was thinking about a Brownian motion that is mirroring, each time being equal to $1$ or $-1$. My hope was, that the quadratic variation is still $t$, such that $(X^2-[X])_{t\ge 0}$ is unbounded. Unfortunately, I am not able to write it down properly.
I would really appreciate help on this problem. Thank you in advance!