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

Abstract

We study the properties of the exponential functional +∞ 0 e −X ↑ (t) dt where X ↑ is a spectrally one-sided Lévy process conditioned to stay positive. In particular, we study finiteness, self-decomposability, existence of finite exponential moments, asymptotic tail at 0 and smoothness of the density.

Almost Sure Behavior for the Local Time of a Diffusion in a Spectrally Negative Lévy Environment

“…In Subsection 4.1 it is precised how V ↑ and V ↑ are defined rigorously. These functionals are studied by the author in [13] where it is proved in Theorems 1.1 and 1.13 that they are indeed finite and well-defined. Let G 1 and G 2 be two independent random variables with G 1…”

“…In [13], the left tail of I(V ↑ ) is linked to the asymptotic behavior of Ψ V . This asymptotic behavior is usually quantified thanks to two real numbers σ and β :…”

“…Note that when Q, the brownian component of V , is positive, then Ψ V has 2-regular variation, and when Q = 0, 1 ≤ σ ≤ β ≤ 2. The asymptotic behavior of P(I(V ↑ ) ≤ x) as x goes to 0 is given by the following theorem from [13] :…”

Asymptotics for Exponential Functionals of Random Walks

“…It is analogous to random walks that I t (ξ) → I ∞ (ξ) a.s. as t → ∞ with I ∞ (ξ) < ∞ a.s. if and only if ξ drifts to infinity. Readers may refer to [10,15,28,29,31] and references therein for many interesting results of I t (ξ) and I ∞ (ξ). In the case I ∞ (ξ) = ∞ a.s., we are usually interested in the decay rate of the following expectation E[F (I t (ξ))] defined as in (1.2), because of its close connection to the long-term properties of random processes in random environment, e.g.…”

Asymptotic Results for Heavy-tailed Lévy Processes and their Exponential Functionals

“…In this case, the characterizations and Wiener-Hopf type factorization of the law of A ∞ (ξ) can be founded in [11,34,35,40]. For more interesting results and properties of A ∞ (ξ), reader may refer to the recent works [6,36,39] and references therein. In the case of A ∞ (ξ) = ∞, we are usually interested in the asymptotic behavior of F (A t (ξ)) for some positive, decreasing function F on (0, ∞) that vanishes at ∞.…”