Why writing $[X,Y]_t$ as $dX_t dY_t$ is so called "abuse of notation"

Question

Why writing $d[X,Y]_t$ as $dX_t dY_t$ or $[B]_t$ as $\int_0^tdt$ is so called "abuse of notation"?

Is it because $[B]_t \rightarrow \int_0^tdt$ a.s. but they are not equal?

Answer 1

The latter is not really an abuse of notation but the former is. Here is why. Suppose you want to compute the Lebesgue-Stieltjes integral $$\int H_s d[X,Y]_s$$ for some process $H$. Now by that notation it may look like there is a double stochastic integral involved here but there is not. The notation is meant as a kind of multiplication rule when computing the quadratic covariation process. The actual "multiplication" rule is as follows. $$[\int M_s dW_s, \int N_s dZ_s] = \int M_sN_s d[W,Z]_s$$ Now let $X$ denote the first term on the LHS and $Y$ the second term. Now we work out $d[X,Y]$ and $dXdY$ (as if we are multiplying).

$$d[X,Y]_s = M_sN_s d[W,Z]_s$$ $$dX_sdY_s = M_sN_s dW_sdZ_s$$

Now I hope the analogy is clear.