Recent developments in exponential functionals of L\'evy processes

TL;DR

This survey reviews twenty years of progress on exponential functionals of Le9vy processes, highlighting structural insights, applications, and recent advances in understanding these complex random variables.

Contribution

It provides a comprehensive overview of recent developments, unifies various results, and emphasizes the probabilistic and analytical techniques used in the field.

Findings

Advances in understanding the structure of exponential functionals

Development of applications across modern probability contexts

Integration of recent results into a unified framework

Abstract

This survey aims to review two decades of progress on exponential functionals of (possibly killed) real-valued L\'evy 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.