I have a question regarding Ito Isometry (https://en.wikipedia.org/wiki/It%C3%B4_isometry). What conditions do I need for the following statement to hold: $$ \mathbb{E}\left[ | \int_0^t K(t,s)\sigma(X_s)dB_s |^2\right] = \mathbb{E}\left[ \int_0^t |K(t,s)\sigma(X_s)|^2ds \right].$$ SInce I dont need any martingale property, I should be fine with integrability conditions on $\sigma,K$, is that correct? Or am I getting problems with the adaptedness? Thx for your help!
2026-03-26 03:19:09.1774495149
Conditions for applying Ito Isometry
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A general Itô is proved here Quadratic Variations and the Ito Isometry
where you also need predictable as mentioned in the comments.