Suppose I have two continuous semimartingales $X$ and $Y$. By Ito's formula for semimartingales, we have
$$ X_tY_t=X_0Y_0+\int_0^t X_sdY_s+\int_0^tY_sdX_s+\langle X,Y\rangle_t $$ where $\langle X,Y\rangle_t$ is the predictable cross-variation between $X$ and $Y$.
The definition of $\langle X,Y\rangle_t$ that I am working with is the adapted process such that $X_tY_t-\langle X,Y\rangle_t$ is a martingale.
However, if $dX_s$ or $dY_s$ contains $ds$ terms, wouldn't it cause $X_tY_t-\langle X,Y\rangle_t$ in the equation above to have a drift, and thus not a martingale anymore?
I only know that $\langle X,X\rangle_t$ is the unique increasing predictable process s.t. $M^2_t-\langle X,X\rangle_t$ is a local martingale where $X_t=M_t+A_t$ is the semimartingale decompositon of $X$.
The covariation $\langle X,Y\rangle_t$ is obtained from polarization.
To make a long story short: when $X,Y$ are continuous semimartingales then in $\langle X,Y\rangle_t$ every finite variation part is filtered out as if it was never there.
Using the historic definition of quadratic variation $$ \langle X,X\rangle_t=\lim_{\max\limits_{i=1,...,n}|t_{i+1}-t_i|\to 0}\sum_{i=1}^n(X_{t_i}-X_{t_{i-1}})^2 $$ it is an instructive exercise to show directly that for an Ito process $$ \textstyle X_t=X_0+\int_0^t\alpha_s\,ds+\int_0^t\beta_s\,dB_s $$ there is $$ \textstyle\langle X,X\rangle_t=\int_0^t\beta_s^2\,ds\,. $$