Give a direct proof for the identity [begin{aligned}mathbb{E} & {left[left(sum_{l=1}^{N} gleft(B_{t_{l-1}}ight)left[left(B_{t_{l}}-B_{t_{l-1}}ight)^{2}-left(t_{l}-t_{l-1}ight)ight]ight)^{2}ight] } & =mathbb{E}left[sum_{l=1}^{N}left|gleft(B_{t_{l-1}}ight)ight|^{2}left[left(B_{t_{l}}-B_{t_{l-1}}ight)^{2}-left(t_{l}-t_{l-1}ight)ight]^{2}ight] .end{aligned}] appearing in
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Give a direct proof for the identity \[\begin{aligned}\mathbb{E} & {\left[\left(\sum_{l=1}^{N} g\left(B_{t_{l-1}}ight)\left[\left(B_{t_{l}}-B_{t_{l-1}}ight)^{2}-\left(t_{l}-t_{l-1}ight)ight]ight)^{2}ight] } \\& =\mathbb{E}\left[\sum_{l=1}^{N}\left|g\left(B_{t_{l-1}}ight)ight|^{2}\left[\left(B_{t_{l}}-B_{t_{l-1}}ight)^{2}-\left(t_{l}-t_{l-1}ight)ight]^{2}ight] .\end{aligned}\]
appearing in the proof of Lemma 18.4.
Data From Leema 18.4
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Related Book For
Brownian Motion A Guide To Random Processes And Stochastic Calculus De Gruyter Textbook
ISBN: 9783110741254
3rd Edition
Authors: René L. Schilling, Björn Böttcher
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