Itô isometry states that $$ \int_0^t {X_sdW_s} \sim \mathcal{N}\left(0, \int_0^t {X_s^2ds}\right) $$ and my question is what is distribution of the process $I_T = \int_t^T {X_sdW_s}$? Is it maybe $$ \int_t^T {X_sdW_s} \sim \mathcal{N}\left(0,\int_t^T {X_s^2ds} \right)? $$
2026-03-27 22:20:15.1774650015
Itô isometry for process that is not starting at 0?
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HINT
$$ \int_t^T X_s dW_s = \int_0^T X_s dW_s - \int_0^t X_s dW_s $$ and you know the distributions of them both, so the end result is clearly normal with zero mean. Can you figure out the variance?