On Dufresne's Perpetuity, Translated and Reflected

Let B(µ) denote a Brownian motion with drift µ. In this paper we study two perpetual integral functionals of B(µ). The first one, introduced and investigated by Dufresne in [5], is$$\int_0^\infty \exp (2B_s^{(\mu)})ds\,,\quad \mu < 0\,.$$It is known that this functional is identical in law with the first hitting time of 0 for a Bessel process with index µ. In particular, we analyze the following reflected (or one-sided) variants of Dufresne's functional$$\int_0^\infty \exp (2B_s^{(\mu)}) {\bf 1}_{\{B_s^{(\mu)} > 0\}} ds\,,$$and$$\int_0^\infty \exp (2B_s^{(\mu)}) {\bf 1}_{\{B_s^{(\mu)} > 0\}} ds\,.$$We shall show in this paper how these functionals can also be connected to hitting times. Our second functional, which we call Dufresne's translated functional, is$${\widehat D}_c^{(\nu)} := \int_0^\infty (c + \exp (B_s^{(\nu)}))^{-2} ds\,, $$where c and ν are positive. This functional has all its moments finite, in contrast to Dufresne's functional which has only some finite moments. We compute explicitly the Laplace transform of ${\widehat D}_c^{(\nu)}$ in the case ν = 1/2 (other parameter values do not seem to allow explicit solutions) and connect this variable, as well as its reflected variants, to hitting times.