Abstract:  We present integration by parts formulas for the joint law of a stable process and its maximum. The argument is based on a multi-level representation for the joint law which uses the theory of convex majorants for stable processes and the Chambers-Mallows-Stuck representation for stable random variables. As applications, we obtain regularity results for the joint law and upper bounds for the density and its space derivatives.

This is joint work with Jorge Gonz\'{a}lez C\'{a}zares, Arturo and Aleksandar Mijatovi\'{c}