Joint law of two stochastic integrals with respect to the same Brownian motion

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?