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

TL;DR

This paper establishes nearly optimal polynomial upper bounds and proves infinite differentiability for the joint density of a stable process and its supremum at a fixed time, using innovative simulation and approximation techniques.

Contribution

It introduces a novel combination of simulation ideas to derive upper bounds and regularity results for the joint density of stable processes and their suprema.

Findings

Established nearly optimal polynomial upper bounds for the joint density.

Proved infinite differentiability of the joint density.

Developed an efficient approximation method using multilevel Monte Carlo.

Abstract

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