Some perpetual integral functionals of the three-dimensional Bessel process

Abstract

We compute the Laplace transforms of some integral functionals of the three-dimensional Bessel process in terms of modified Bessel functions, Gauss’ hypergeometric functions, and confluent hypergeometric functions. Some new results are obtained, and several established results, such as Dufresne’s perpetuity and a particular case of its translated version, are recovered. In particular, we derive the Laplace transform of the two-dimensional random variable [Formula: see text] where [Formula: see text] is a three-dimensional Bessel process and [Formula: see text]. A crucial step of the derivation is to construct a martingale, as performed in the Bass solution of the Skorokhod embedding problem, with respect to an enlarged filtration.