Quadratic variation of a square integrable continuous martingale

Question

Is the total variation of the quadratic variation process of a square integrable right continuous martingale bounded?

Answer 1

Since the quadratic variation process is non-decreasing, its total variation is equal to its value and the quadratic variation does not need to be bounded, although it will have finite expectation and hence be a.s. finite. For example, let $B$ be a standard Brownian motion, $\tau := \inf\{t : |B_t| = 1\}$, and $M_t := B_{t \wedge \tau}$. Then $M$ is square integrable (in fact $M$ is bounded) and continuous, but $\langle M,M \rangle_t = t \wedge \tau$ and $\tau$ is not bounded.