Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes
The authors study exponential functionals of hypergeometric Lévy processes which include, for instance, the Lamperi-stable processes. The Mellin transform as well as convergent and asymptotic series representations are obtained for the density of the exponential functional. The proof relies on a verification result which gives explicit conditions to identify the Mellin transform uniquely. The results are applied to fluctuations of stable processes. In particular, they determine densities of the supremum of stable processes, of the entrance law of the excursion measure of a stable processes reflected at its past infimum, and of the entrance law of a stable process conditioned to stay positive. Further results on multidimensional symmetric stable processes are obtained.
convergent and asymptotic series expansion
scientific article