A Note on Some New Perpetuities

In a recent paper, Salminen and Yor relate the distribution of the Dufresne's reflected perpetuity where Bµ is Brownian motion with drift µ>0, to the hitting time of a reflected Bessel process. In this contribution, we adapt the results of Salminen and Yor in several ways. First, we use spectral theory to obtain a series expansion for the distribution of I+ that renders this quantity applicable to actuarial purposes. We also study the exponential functionals where Xµa is a skew Brownian motion with drift µ>0.