Let $(X_t)_{t\geq 0}$ be a continuous process given by $$ X_t:=\int_0^tb(\omega,s)\,dB_s-\int_0^tb(\omega,s)^2\,ds $$ where $b\in L^2_{\mathrm{loc}}$ such that $\int_0^tb^2\,ds\to\infty$ with probability one. Then, I want to find the law of $\sup_{t\geq 0}X_t$.
I don't know how to start with this problem. I tried consider a simpler case where $X_t = B_t - t$, but I still don't know what to do. I think this has something to do with that we can write $X_t = M_t - \langle M\rangle_t$ where $M_t$ is a local martingale. I'd appreciate any help, thank you!