I am having some trouble working out $E\left[ \int_0^tg^2(\tau)dW_\tau \right]$. I don't even really know where to begin, I'm very new to Ito Calculus and any help or nudge in the right direction would be much appreciated.
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Is $g$ deterministic, or another random process? Either way, it's probably useful to know that Ito integrals are local martingales. – user6247850 Dec 08 '22 at 23:52
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It might be useful to read: https://math.stackexchange.com/questions/232932/it%C5%8D-integral-has-expectation-zero – openspace Dec 09 '22 at 10:21