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.


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