Let $a:\mathbb R_+\to [1,2]$ be "smooth". Given a standard Brownian motion $W$, define for $t\ge 0$
$$X_t:=\int_0^t\frac{1}{a(s)}dW_s \quad \mbox{and}\quad Y_t:=\sup_{0\le u\le t} \int_0^u a(s)dW_s.$$
What is the joint law of $(X_t,Y_t)$? More precisely, for any $z>0$, is there any closed form expression for the expectation $\mathbb E\left[\exp(-X_t){\bf 1}_{\{Y_t<z\}}\right]$?
Any answer, comments or references are highly appreciated.
