Bivariate Bernstein-gamma functions, potential measures, and asymptotics of exponential functionals of Lévy processes

Abstract

Let $\xi$ be a Lévy process and $I_{\xi }(t):={\int }_0^t{e}^{-{\xi }_s}\mathrm{d}s$, $t\ge 0,$ be the exponential functional of Lévy 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 $\mathbb{E}\left(I_{\xi }^{-a}(t)1_{\left\{I_{\xi }(t)\le x\right\}}\right)$, $a\in (0,1)$, $x\in (0,\infty ],$ 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.

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