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I'm interested in knowing if there are any results of the following type:

Let $(B_t^i,\mathcal F_t^i)$, $i=1,2$ be two Brownian motions. Suppose the filtration $\{\mathcal F_t^1\}$ is contained in the filtration $\{\mathcal F_t^2\}$. As a result, $B^2$ may not be adapted to $\{\mathcal F_t^1\}$, while $B^1$ may not be a semi-martingale with respect to $\{\mathcal F_t^2\}$.

  1. Is there a way to define (or make sense of ) the quadratic covariation process $[B^1,B^2]$ ?
  2. Given a sequence of refining partitions $\{\pi_n\}$ with $|\pi_n| \to 0$, does the Riemann sum $\Sigma_{t_i \in \pi_n,t_i\leq t} (B_{t_{i+1}}^1-B_{t_i}^1)(B_{t_{i+1}}^2-B_{t_i}^2)$ converge in an sense?(say in probability,etc)

I have looked up things like Young integral and Follmer's pathwise integral and none of those seem to be applicable in this case. Any help and suggestion would be greatly appreciated!

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