Prove that the density of the pair (left(exp left(W_{Theta}+ u Theta ight), A_{Theta}^{( u)} ight)), where (A_{t}^{(

Prove that the density of the pair \(\left(\exp \left(W_{\Theta}+\nu \Theta\right), A_{\Theta}^{(\nu)}\right)\), where \(A_{t}^{(\nu)}=\int_{0}^{t} e^{2\left(W_{s}+\nu s\right)} d s\), is

\[\frac{\theta^{2}}{2 x^{2+\lambda}} x^{\nu} p_{a}^{(\lambda)}(1, x) \mathbb{1}_{\{x>0\}} \mathbb{1}_{\{a>0\}} d x d a \]

with \(\lambda^{2}=\theta^{2}+\nu^{2}\).