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Given a filtered space $(\Omega,\mathcal F,\mathbb F,\mathbb P)$ supporting a Brownian Motion $B$, where the filtration $\mathcal F$ is the augmented Brownian filtration, the Azema's martingale is defined by $M_t=\mathbb E(B_t|\mathcal G_t)$, where $\mathcal G_t=\sigma(sign(B_s):s\leq t)$, completed over all $\mathbb P$ null sets. It can be shown that the filtration generated by $M$ is exactly $\mathbb G$, so that $M$ is a martingale under its own filtration. It can also be shown that $M$ is not an $\mathbb F$-semimartingale. My question is, is the quadratic covariation between $M$ and $B$ well-defined? That is, given a sequence of partition $\pi_n$ with $|\pi_n|$ goes to zero, does $\sum_{\pi_n\cap[0,t]}(M_{t_i}-M_{t_{i-1}})(B_{t_i}-B_{t_{i-1}})$ converges in any sense?

Any advice and help are greatly appreciated!

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The Riemann sum converges to zero in probability since $M$ is quadratic pure jump and $B$ is continuous.

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