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

Abstract

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 diffusions on the hyperbolic halfplane. Emphasis is put on the fact that the results are obtained in two independent, very different fashions (invariant diffusions on the hyperbolic halfplane and Bessel processes).