Density of stochastic integral

Question

I am working on finding the PDF of $X_t^2$,

where $X_t = \int_0^t A(u) \,dW_u$, a Wiener integral, i.e., $W_t$ is Brownian motion and $A(t)$ is a deterministic function.

Here, would like to ask that do you guys have any idea about what does the PDF of $X_t^2$ look like, or how to find it ?

Thank you in advance.

Answer 1

You can determine the sde that $X_t^2$ satisfies using Ito's lemma. Then you can find the pdf by solving the forward Kolmogorov equations (Fokker-Planck).