Exponential functionals of spectrally one-sided l{\'e}vy processes conditioned to stay positive

TL;DR

This paper investigates the mathematical properties of exponential functionals of spectrally one-sided Lévy processes conditioned to stay positive, focusing on their finiteness, distributional features, and tail behavior.

Contribution

It provides new insights into the distributional characteristics and tail asymptotics of exponential functionals for conditioned Lévy processes, expanding understanding in stochastic process theory.

Findings

Finiteness conditions established for the exponential functional.

Proved self-decomposability of the distribution.

Derived asymptotic tail behavior at zero.

Abstract

We study the properties of the exponential functional $\int\_0^{+ \infty} e^{- X^{\uparrow} (t)}dt$ where $X^{\uparrow}$ is a spectrally one-sided L{\'e}vy 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.