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The integration by parts formula states: If $X$ and $Y$ are continuous semimartingales, then

$$\int_0^t Y_s dX_s = X_t Y_t - X_0 Y_0 - \int_0^t X_s dY_s - [X,Y]_t,$$

for all $t \geq 0$. My question is now why the covariance terms vanishes for functions of bounded variation. Thanks in advance.

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Consider the polarization identity for covariance given by $ \langle X,Y \rangle = \frac{1}{4} (\langle X+Y \rangle -\langle X-Y \rangle ) $. You can show that both X+Y and X-Y also have finite variation (use the definition of variation and then triangle inequality) so then both terms in the polarisation identity vanish as finite variation implies zero quadratic variation. Hope that helps

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