Non-symmetric hitting distributions on the hyperbolic half-plane and subordinated perpetuities

We study the law of functionals whose prototype is $\int_0^{+\infty}$ $e^{B{_s}^{(\nu)}} dW{_s}{^{(\mu)}}$, where $B^{(\nu)}$, $W^{(\mu)}$ are independent Brownian motions with drift. These functionals appear naturally in risk theory as well as in the study of invariant di ffusions on the hyperbolic half plane. Emphasis is put on the fact that the results are obtained in two independent , very diff erent fashions ( invariant di ffusions on the hyperbolic half plane and Bessel processes).