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.
"On extrema of stable processes." Ann. Probab. 39 (3) 1027 - 1060, May 2011. https://doi.org/10.1214/10-AOP577