A Multi level method to study the law of the supremum of a stable process

Thời gian: 14h Thứ 5, ngày 19/10/2023

Địa điểm: Phòng 507 nhà A6

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Tóm tắt: There are very few methods to study the law of the supremum of a jump process using Malliavin Calculus. One of them was proposed by Bouleau and Denis using the lent particle method. This method could be used to obtain the existence of the law. In this presentation we are interested in obtaining the regularity and optimal estimates of the joint law of the stable process and its supremum. Clearly the problems are not only related to the differentiability of the functional given by the supremum process but also related to the restricted amount of nite moments that the stable process has.

We used a combination of three effective simulation methods for the maximum of the stable process together with an interpolation methodology (similar to one introduced to V. Bally and L. Caramellino) to obtain almost optimal upper bounds for the joint law of the stable law and its supremum:

1. The convex majorant approach to supremum of Levy processes
2. The Chambers-Mallows-Stuck simulation method for stable laws
3. The Multi Level Monte Carlo method

This is joint work with Jorge Gonzalez-Cazares and Alex Mijatovic (Warwick University and Turing Institute).