What is $\int_{0}^{t}\sqrt{\frac{1}{T-s}}dW(s)$ where $0\le t <T$ and $T \in \mathbb{R}$ and W is a one dimensional brownian motion.
I tried applying Ito's rule with $f(x)= \sqrt{T-x}$. but I did not get the answer.
2026-03-29 19:49:50.1774813790
$\int_{0}^{t}\sqrt{\frac{1}{T-s}}dW(s)$
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Hint
Yes, Ito's rule won't help to my knowledge. That's useful for when you have $W$'s in the integrand. You're actually in the somewhat simpler territory where you don't. Think about it: the $dW_s's$ are independent normals with mean zero and variance $ds$ and they are multiplied by $\frac{1}{\sqrt{T-s}}.$ So you are adding up a bunch of independent normals with mean zero and variance $\frac{ds}{T-s}.$ What do you get?