Joint density of a stable process and its supremum: regularity and upper bounds

Jorge González Cázares, Arturo Kohatsu-Higa, Aleksandar Mijatović

Published 2020-08-05Version 1

This article develops integration-by-parts formulae for the joint law of a stable process and its supremum at a fixed time. The argument rests on a multilevel representation for the joint law and uses ideas from Malliavin calculus, the theory of convex majorants for stable processes and the Chambers-Mallows-Stuck representation for stable laws. As our main application, we prove that an infinitely differentiable joint density exists and establish upper bounds (on the entire support of the joint law) for this density and its partial derivatives of any order.