Quadratic Variation

Question

Let $$ [X,Y]=\int_{0}^{\tau} (dX_t)(dY_t)$$ where $X_t$ is cadlag process and $Y_t$ is a Brownian motion. Can we say $$ \frac{\partial}{\partial\tau}\int_{0}^{\tau} (dX_t)(dY_t)= 0 $$

Answer 1

No. Take $X_t=Y_t=B_t$ to be a Brownian motion started from zero. Then $[X,Y]_t=t$, so in this case $\frac{d}{dt}[X,Y]_t=1$, not zero.