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

We study the law of functionals whose prototype is integral(0)(+infinity) e(s)(B(V)) 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 diffusions on the hyperbolic half-plane. Emphasis is put on the fact that the results-are obtained in two independent, very different fashions (invariant diffusions on the hyperbolic half-plane and Bessel processes).