On the density of exponential functionals of Lévy processes

Abstract

In this paper, we study the existence of the density associated to the exponential functional of the Lévy process ξ,where e q is an independent exponential r.v. with parameter q ≥ 0. In the case when ξ is the negative of a subordinator, we prove that the density of I eq , here denoted by k, satisfies an integral equation that generalizes the one found by Carmona et al. [7]. Finally when q = 0, we describe explicitly the asymptotic behaviour at 0 of the density k when ξ is the negative of a subordinator and at ∞ when ξ is a spectrally positive Lévy process that drifts to +∞.

Asymptotic behaviour of exponential functionals of Lévy processes with applications to random processes in random environment

“…Most of the known results on I ∞ (ξ) and I eq (ξ) are related to the knowledge of their densities or the behaviour of their tail distributions. According to Theorem 3.9 in Bertoin et al [9], there exists a density for I ∞ (ξ), here denoted by h. In the case when q > 0, the existence of the density of I eq (ξ) appears in Pardo et al [30]. Moreover, according to Theorem 2.2. in Kuznetsov et al [24], under the assumption that E[|ξ 1 |] < ∞, the density h is completely determined by the following integral equation: for v > 0, We refer to [9,10,24,30], and the references therein, for more details about these facts.…”

“…There is a vast literature about exponential functionals of Lévy processes drifting to +∞ or killed at an independent exponential time e q with parameter q ≥ 0, see for instance [9,10,30]. For a Lévy process ξ satisfying one of these assumptions, I ∞ (ξ) or I eq (ξ) is finite almost surely with an absolute continuous density.…”

Exponential functionals of spectrally one-sided Lévy processes conditioned to stay positive

“…Our last result refines (1.9) when Y is spectrally positive and π satisfies (A). It complements the study [21] of densities of exponential functionals of spectrally positive Lévy processes and we prove it as a consequence of the factorization identity in [16] together with a result of [19] about subordinators. Proposition 1.17.…”

“…For example they are fundamental in the study of diffusions in random environments and appear in many applications such as the study of self-similar Markov processes and mathematical finance, see Bertoin, Yor [6] and Pardo, Rivero [18] for surveys on those functionals and their applications. For a general Lévy process, equivalent conditions for the finiteness of the exponential functional are given in [6], the asymptotic tail at +∞ of the functional is studied in [26], [13], [27], [28] (see also [6], [18]), the absolute continuity is proved in [11], [19], and properties of the density (such as regularity) are studied in [14], [19], [23]. Recently, factorization identities for exponential functionals of Lévy processes have been proved in [16], [22], [23] (see also [18]).…”

Exponential functionals of Markov additive processes

“…Due to this connection, the resulting importance in applications, and their complexity, exponential functionals have gained a lot of attention from various researchers over the last decades, see e.g. [5,6,8,20,22,23] to name just a few.…”