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This article uses a combination of three ideas from simulation to establish anearly optimal polynomial upper bound for the joint density of the stableprocess and its associated supremum at a fixed time on the entire support ofthe joint law. The representation of the concave majorant of the stable processand the Chambers-Mallows-Stuck representation for stable laws are used todefine an approximation of the random vector of interest. An interpolationtechnique using multilevel Monte Carlo is applied to accelerate theapproximation, allowing us to establish the infinite differentiability of thejoint density as well as nearly optimal polynomial upper bounds for the jointmixed derivatives of any order.

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