I have a question about calculating covariances of local martingales. Suppose $M_1$ and $M_2$ are local martingales. Put $M = M_1+M_2$. Is there a nice way to calculation $[M]$ in terms of $[M_1]$ and $[M_2]$?
I feel that if $B_1$ and $B_2$ are independent Brownian motion, and $M_1 = \int f(B_1(s),B_2(s),s) \, dB_1(s)$ and $M_2 = \int g(B_1(s),B_2(s),s) \, dB_2(s)$, then $[M] = \int f^2 + g^2 \, dt$.