Bivariate Bernstein-gamma functions, potential measures, and asymptotics of exponential functionals of L\'evy processes

TL;DR

This paper analyzes the asymptotic behavior of exponential functionals of Lévy processes, providing new bounds, convergence results, and a novel approach using Mellin inversion and Bernstein-gamma functions, advancing the theoretical understanding of these stochastic processes.

Contribution

It introduces a new methodology combining Mellin inversion and Bernstein-gamma functions to study exponential functionals of Lévy processes, with generalizations and improved results over existing literature.

Findings

Derived upper bounds on decay rates of expected functionals

Established weak convergence of rescaled exponential functionals

Linked Bernstein-gamma functions to q-potentials of Lévy processes

Abstract

Let $\xi$ be a L\'{e}vy process and $I_\xi(t):=\int_{0}^te^{-\xi_s}\mathrm{d} s$, $t\geq 0,$ be the exponential functional of L\'{e}vy processes on deterministic horizon. Given that $\lim_{t\to \infty}\xi_t=-\infty$ we evaluate for general functions $F$ an upper bound on the rate of decay of $\mathbb{E}\left(F(I_\xi(t))\right)$ based on an explicit integral criterion. When $\mathbb{E}\left(\xi_1\right)\in\left(-\infty,0\right)$ and $\mathbb{P}\left(\xi_1>t\right)$ is regularly varying of index $\alpha>1$ at infinity, we show that the law of $I_\xi(t)$, suitably normed and rescaled, converges weakly to a probability measure stemming from a new generalisation of the product factorisation of classical exponential functionals. These results substantially improve upon the existing literature and are obtained via a novel combination between Mellin inversion of the Laplace transform of $\mathbb{E}\left(I^{-a}_{\xi}(t)\mathbf{1}_{\left\{I_{\xi}(t)\leq x\right\}}\right)$, $a\in (0,1)$, $x\in(0,\infty],$ and Tauberian theory augmented for integer-valued $\alpha$ by a suitable application of the one-large jump principle in the context of the de Haan theory. The methodology rests upon the representation of the aforementioned Mellin transform in terms of the recently introduced bivariate Bernstein-gamma functions for which we develop the following new results of independent interest (for general $\xi$): we link these functions to the $q$-potentials of $\xi$; we show that their derivatives at zero are finite upon the finiteness of the aforementioned integral criterion; we offer neat estimates of those derivatives along complex lines. These results are useful in various applications of the exponential functionals themselves and in different contexts where properties of bivariate Bernstein-gamma functions are needed. $\xi$ need not be non-lattice.