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Kunita-Watanabe identity:

Let $M,N$ be local martingales, $H$ be a locally bounded previsible process, then $[H\cdot M,N]=H\cdot[M,N],$ where $[M,N]$ is covariation.

I am going though the proof, but 1 step is omitted. I am trying to use the polarization to conclude that $[H\cdot M,N]+[M,H\cdot N]=2H\cdot [M,N],$ i.e. $\frac{1}{4}([H\cdot M+N]-[H\cdot M-N]+[M+H\cdot N]-[M-H\cdot N])=2H\cdot [M,N],(*)$ where $[X]$ denotes quadratic variation of $X$. How can I show $(*)$?

Thank you.

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    I recommend using the proof [here](http://www.statslab.cam.ac.uk/~arnab/SC/sc.pdf) instead. To show $(*)$, you start by noting that $2H=\frac{(H+1)^2-(H-1)^2}{2}$.2012-05-21

0 Answers 0