Tricky question about Ito's stochastic integral and continal law 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.

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 $B$.