Let and be two martingales, . Show that
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![\displaystyle \sum_{k=2}^{n}E(\theta_{k}-\theta_{k-1})(\phi_{k}-\phi_{k-1}) \\ = E\sum_{k=2}^{n}E\left[(\theta_{k}-\theta_{k-1})(\phi_{k}-\phi_{k-1})|\mathcal{D}_{k-1}\right]\mbox{ by total probability formula} \\ = E\sum_{k=2}^{n}E(\theta_{k}\phi_{k}|\mathcal{D}_{k-1})-E(\theta_{k}\phi_{k-1}|\mathcal{D}_{k-1})-E(\theta_{k-1}\phi_{k}|\mathcal{D}_{k-1})+E(\theta_{k-1}\phi_{k-1}|\mathcal{D}_{k-1}) \\ = E\sum_{k=2}^{n}E(\theta_{k}\phi_{k}|\mathcal{D}_{k-1})-\phi_{k-1}E(\theta_{k}|\mathcal{D}_{k-1})-\theta_{k-1}E(\phi_{k}|\mathcal{D}_{k-1})+E(\theta_{k-1}\phi_{k-1}|\mathcal{D}_{k-1}) \\ \mbox{ because }\phi_{k-1},\theta_{k-1}\mbox{ are }\mathcal{D}_{k-1}\mbox{-measurable} \\ = E\sum_{k=2}^{n}E(\theta_{k}\phi_{k}|\mathcal{D}_{k-1})-2\phi_{k-1}\theta_{k-1}+E(\theta_{k-1}\phi_{k-1}|\mathcal{D}_{k-1})\mbox{ by definition of martingale} \\ = E\sum_{k=2}^{n}E(\theta_{k}\phi_{k}|\mathcal{D}_{k-1})-\phi_{k-1}\theta_{k-1}\mbox{ because }\theta_{k-1}\phi_{k-1}\mbox{ is }\mathcal{D}_{k-1}\mbox{-measurable} \\ = E\sum_{k=2}^{n}E(\theta_{k}\phi_{k}|\mathcal{D}_{k-1})-E(\phi_{k-1}\theta_{k-1}|\mathcal{D}_{k-1}) \\ = E\theta_{n}\phi_{n}-E\phi_{1}\theta_{1} \\ = E\theta_{n}\phi_{n}\mbox{ since }\theta_{1}=\phi_{1}=0](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Csum_%7Bk%3D2%7D%5E%7Bn%7DE%28%5Ctheta_%7Bk%7D-%5Ctheta_%7Bk-1%7D%29%28%5Cphi_%7Bk%7D-%5Cphi_%7Bk-1%7D%29+%5C%5C+%3D+E%5Csum_%7Bk%3D2%7D%5E%7Bn%7DE%5Cleft%5B%28%5Ctheta_%7Bk%7D-%5Ctheta_%7Bk-1%7D%29%28%5Cphi_%7Bk%7D-%5Cphi_%7Bk-1%7D%29%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%5Cright%5D%5Cmbox%7B+by+total+probability+formula%7D+%5C%5C+%3D+E%5Csum_%7Bk%3D2%7D%5E%7Bn%7DE%28%5Ctheta_%7Bk%7D%5Cphi_%7Bk%7D%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%29-E%28%5Ctheta_%7Bk%7D%5Cphi_%7Bk-1%7D%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%29-E%28%5Ctheta_%7Bk-1%7D%5Cphi_%7Bk%7D%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%29%2BE%28%5Ctheta_%7Bk-1%7D%5Cphi_%7Bk-1%7D%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%29+%5C%5C+%3D+E%5Csum_%7Bk%3D2%7D%5E%7Bn%7DE%28%5Ctheta_%7Bk%7D%5Cphi_%7Bk%7D%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%29-%5Cphi_%7Bk-1%7DE%28%5Ctheta_%7Bk%7D%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%29-%5Ctheta_%7Bk-1%7DE%28%5Cphi_%7Bk%7D%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%29%2BE%28%5Ctheta_%7Bk-1%7D%5Cphi_%7Bk-1%7D%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%29+%5C%5C+%5Cmbox%7B+because+%7D%5Cphi_%7Bk-1%7D%2C%5Ctheta_%7Bk-1%7D%5Cmbox%7B+are+%7D%5Cmathcal%7BD%7D_%7Bk-1%7D%5Cmbox%7B-measurable%7D+%5C%5C+%3D+E%5Csum_%7Bk%3D2%7D%5E%7Bn%7DE%28%5Ctheta_%7Bk%7D%5Cphi_%7Bk%7D%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%29-2%5Cphi_%7Bk-1%7D%5Ctheta_%7Bk-1%7D%2BE%28%5Ctheta_%7Bk-1%7D%5Cphi_%7Bk-1%7D%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%29%5Cmbox%7B+by+definition+of+martingale%7D+%5C%5C+%3D+E%5Csum_%7Bk%3D2%7D%5E%7Bn%7DE%28%5Ctheta_%7Bk%7D%5Cphi_%7Bk%7D%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%29-%5Cphi_%7Bk-1%7D%5Ctheta_%7Bk-1%7D%5Cmbox%7B+because+%7D%5Ctheta_%7Bk-1%7D%5Cphi_%7Bk-1%7D%5Cmbox%7B+is+%7D%5Cmathcal%7BD%7D_%7Bk-1%7D%5Cmbox%7B-measurable%7D+%5C%5C+%3D+E%5Csum_%7Bk%3D2%7D%5E%7Bn%7DE%28%5Ctheta_%7Bk%7D%5Cphi_%7Bk%7D%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%29-E%28%5Cphi_%7Bk-1%7D%5Ctheta_%7Bk-1%7D%7C%5Cmathcal%7BD%7D_%7Bk-1%7D%29+%5C%5C+%3D+E%5Ctheta_%7Bn%7D%5Cphi_%7Bn%7D-E%5Cphi_%7B1%7D%5Ctheta_%7B1%7D+%5C%5C+%3D+E%5Ctheta_%7Bn%7D%5Cphi_%7Bn%7D%5Cmbox%7B+since+%7D%5Ctheta_%7B1%7D%3D%5Cphi_%7B1%7D%3D0+&bg=ffffff&fg=333333&s=0&c=20201002)
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