On Dufresne's Perpetuity, Translated and Reflected

Paavo Salminen and Marc Yor
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Paavo Salminen: Åbo Akademi, Mathematical Department, FIN-20500 Åbo, Finland
Marc Yor: Université Pierre et Marie Curie, Laboratoire de Probabilités, 4, Place Jussieu, Case 188, F-75252 Paris Cedex 05, France

Chapter 16 in Stochastic Processes and Applications to Mathematical Finance, 2004, pp 337-354 from World Scientific Publishing Co. Pte. Ltd.

Abstract: AbstractLet 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.

Keywords: Stochastic Processes; Stochastic Differential Equations; Malliavin Calculus; Stochastic Control and Optimization; Functionals of Brownian Motions and Lévy Processes; Stochastic Models of Financial Market; Derivative Pricing; Hedging Problem (search for similar items in EconPapers)
Date: 2004
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