Recent developments in exponential functionals of Lévy processes
Martin Minchev, Mladen Savov
TL;DR
The survey advances the theory of exponential functionals of Lévy processes by unifying moment recursions, Mellin transforms, and Bernstein-gamma functions within a Wiener–Hopf framework. It provides general formulas for the Mellin transform of $I_\Psi$, explicit solutions in Brownian, hypergeometric, and meromorphic Lévy models, and detailed density, support, and tail analyses. Key contributions include complete Mellin-transform representations, factorisations of $I_\Psi$, and comprehensive asymptotics (large and small) together with integral equations linking the law to Lévy data. The Bernstein-gamma apparatus offers a powerful, broadly applicable toolkit for deriving exact expressions and asymptotics, with clear implications for applications in stochastic processes, self-similar structures, and related areas in probability theory.
Abstract
This survey aims to review two decades of progress on exponential functionals of (possibly killed) real-valued Lévy processes. Since the publication of the seminal survey by Bertoin and Yor, substantial advances have been made in understanding the structure and properties of these random variables. At the same time, numerous applications of these quantities have emerged across various different contexts of modern applied probability. Motivated by all this, in this manuscript, we provide a detailed overview of these developments, beginning with a discussion of the class of special functions that have played a central role in recent progress, and then organising the main results on exponential functionals into thematic groups. Moreover, we complement several of these results and set them within a unified framework. Throughout, we strive to offer a coherent historical account of each contribution, highlighting both the probabilistic and analytical techniques that have driven the advances in the field.