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.
Funding Statement
JGC and AM are supported by EPSRC grant EP/V009478/1 and The Alan Turing Institute under the EPSRC grant EP/N510129/1; AM was supported by the Turing Fellowship funded by the Programme on Data-Centric Engineering of Lloyd’s Register Foundation; AK-H was supported by JSPS KAKENHI Grant Number 20K03666.
Acknowledgements
We would like to thank the anonymous reader for pointing us towards the reference [24], which helped paint a more complete picture of the existing analytical results in the context of joint stable densities.
Citation
Jorge Ignacio González Cázares. Arturo Kohatsu-Higa. Aleksandar Mijatović. "Joint density of the stable process and its supremum: Regularity and upper bounds." Bernoulli 29 (4) 3443 - 3469, November 2023. https://doi.org/10.3150/23-BEJ1590
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