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Consider $B=(B_t)_{t\geq 0}$ real $\mathcal F_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. Then, consider a new real $\mathcal F_t$ - brownian motion $\tilde{B}=(\tilde{B_0})_{t\geq t}$ independent of $B$ as weel as a process $H=(H_t)_{t\geq 0}$ given by

$$ H_t := \frac{1}{\int _0^t f^2(B_s) ~ds}\int _0^t f(B_s) ~d \tilde B_s \mathbf 1_{\{ \int _0^t f^2(B_s) ~ds>0\}}, \ t\geq 0,$$

where $f \in \mathcal C^0(\mathbb R, \mathbb R)$ and $f \not\equiv 0$.

What is the conditional distribution of $H_t$ knowing $B$?

I dont have any idea on how to start to approach it. Any advice will be strongly appreciate. Thank's in advance.

Crossposted on overflow.

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Answered on overflow by The Bridge.

$$ \mathcal L(H_t| B) = \mathcal N (0,1)$$ since the integral which defines $H$ is a Wiener integral if we know the all path of $B$.

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