Celebratio Mathematica — Yor — Complete Bibliography

Complete Bibliography

Works connected to Shinzo Watanabe

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P. Salminen and M. Yor: “On Dufresne’s perpetuity, translated and reflected,” pp. 337–354 in Stochastic processes and applications to mathematical finance (Kusatsu, Japan, 5–9 March 2003). Edited by J. Akahori, S. Ogawa, and S. Watanabe. World Scientific (River Edge, NJ), 2004. MR 2202705 Zbl 1323.60113 incollection

Let \( B(\mu) \) denote a Brownian motion with drift \( \mu \). In this paper we study two perpetual integral functionals of \( B(\mu) \). The first one, introduced and investigated by Dufresne in [1990], is \[ \int_0^{\infty}\exp(2B_s^{(\mu)})\,ds,\quad \mu\lt 0. \] It is known that this functional is identical in law with the first hitting time of 0 for a Bessel process with index \( \mu \). In particular, we analyze the following reflected (or one-sided) variants of Dufresne’s functional \[ \int_0^{\infty}\exp(2B_s^{(\mu)})\,\mathbf{1}_{\{B_s^{(\mu)}\gt 0\}}\,ds, \] and \[ \int_0^{\infty}\exp(2B_s^{(\mu)})\,\mathbf{1}_{\{B_s^{(\mu)}\lt 0\}}\,ds, \] 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 \[ \hat{D}_c^{(\nu)} := \int_0^{\infty}\bigl(c+\exp(B_s^{(\nu)}) \bigr)^{-2}ds, \] where \( c \) and \( \mu \) 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 \( \hat{D}_c^{(\nu)} \) in the case \( \nu = 1/2 \) (other parameter values do not seem to allow explicit solutions) and connect this variable, as well as its reflected variants, to hitting times.