On extrema of stable processes

TL;DR

This paper investigates the extrema of stable processes by deriving explicit formulas and identities for their distribution, Wiener--Hopf factors, and related functions, using connections to elliptic-like functions.

Contribution

It introduces a novel approach linking Wiener--Hopf factors of stable processes to elliptic-like functions, enabling explicit formulas and new identities.

Findings

Explicit series and asymptotic formulas for supremum density

Closed-form expressions for Wiener--Hopf factors

Functional identities and product representations for supremum-related functions

Abstract

We study the Wiener--Hopf factorization and the distribution of extrema for general stable processes. By connecting the Wiener--Hopf factors with a certain elliptic-like function we are able to obtain many explicit and general results, such as infinite series representations and asymptotic expansions for the density of supremum, explicit expressions for the Wiener--Hopf factors and the Mellin transform of the supremum, quasi-periodicity and functional identities for these functions, finite product representations in some special cases and identities in distribution satisfied by the supremum functional.