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It is known that stochastic integral must satisfy the isometry property which is $$ \mathbb{E}\left[ \left( \int_0^T X_t~dB_t\right)^2 \right] = \mathbb{E} \left[ \int_0^T X^2_t~dt \right] . $$ I am trying to prove this property for a simple stochastic process. What I said so far that is $$ \mathbb{E}\left[\sum_{i=0}^{n-1} X_i \left(B(t_{i+1})-B(t_i)\right)\right]^2, $$ then I am stuck. I know that we should to write the square sum as double sum to continue the proof but I couldn't do it. Any help please!

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    I don't understand the sentence after "What I said so far is..."2012-07-04

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