According to the Wikipedia page on Itô calculus, the equation for integration by parts is
$$X_tY_t = X_0Y_0 + \int_0^t X_{s-} d Y_s + \int_0^t Y_{s-} d X_s + [X,Y]_t.$$ Is there a missing term $-[X,Y]_0$ here? Because when I insert $t=0$, it gives us that $$X_0Y_0 = X_0Y_0 + \int_0^0 X_{s-} d Y_s + \int_0^0 Y_{s-} d X_s + [X,Y]_0,$$ hence $$0=[X,Y]_0.$$ I don't see why that would be true in general, especially when $[X,Y]_t$ is nonzero.
