Any idea on solving the joint distribution of $X_T=\int_0^T \alpha_t dZ_t$ and $Y_T=\int_0^T \alpha_t^2 dt$ ? Here $X_T$ is an Ito integral and $Z_t$ is a standard Brownian process. When $\alpha_t$ and $Z_t$ are independent, it's easy to show $X_T$ is a conditional normal random variable given $Y_T$, but it's not the case when $\alpha_t$ and $Z_t$ are correlated.



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