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!



Azema's martingale and quadratic covariation
The Riemann sum converges to zero in probability since $M$ is quadratic pure jump and $B$ is continuous.


