Let $M_t=\int_0^tX_udB_u$, where $X$ is progressive process. Then is it true that $M_1$ and $B_{\int_0^1X^2_udu}$ have same distribution?
It is true when $X$ is deterministic, but can this be true when $X$ is random?
If true I am assuming one would employ Dubins-Schwarz Theorem to show, but I am not sure how to do it.
In particular, Dubins-Schwartz says there exists some BM $W$ such that $M_t=W_{\int_0^t X_udu}$, but the joint distribution $(M,B)$ can be different from $(M,W)$?
$M_{1}$ is centered Gaussian with variance $\int_{0}^{1}X_{u}^{2}du$. The same holds for $B_{\int_{0}^{1}X_{u}^{2}du}$.