ON EXTREMA OF STABLE PROCESSES

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.