The Library
The harmonic extension technique with applications to optimal stopping
Tools
Herman, John Andrew (2020) The harmonic extension technique with applications to optimal stopping. PhD thesis, University of Warwick.
|
PDF
WRAP_Theses_Herman_2020.pdf - Submitted Version - Requires a PDF viewer. Download (3127Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b3492800~S15
Abstract
In this thesis we investigate the harmonic extension method first popularised by Caffarelli & Silvestre in [14] which allows the fractional Laplacian to be represented in terms of data retrieved from the solution uf to a local PDE problem. We generalise this method to obtain local representations for a family of non-local operators −ψ(−Lx) where ψ is a complete Bernstein function and Lx is the generator of a diffusion semigroup on some Banach space using two different approaches; one based upon stochastic analysis and the other based upon semigroup theory. Underlying both of these approaches is the Krein correspondence which gives a one-to-one correspondence between complete Bernstein functions and a family of functions known as Krein strings. We study this correspondence and focus on a particular function ϕλ called the the extension function which provides the key to understanding the extension method.
As an application of this method, we show how an obstacle problem associated with the non-local operator −ψ(−Lx) can be studied via the techniques found in [9] which can usually only be applied to local problems. Under certain conditions placed on Lx and the obstacle G, we show that the solution V to this problem lies in the L 2 -domain of the operator −ψ(−Lx). Furthermore, if ψ arises as the Laplace exponent of the inverse local time of a one-dimensional diffusion process, then we show that the solution will belong to the L p -domain of the operator −ψ(−Lx) allowing us to prove a regularity result for V .
Item Type: | Thesis (PhD) | ||||
---|---|---|---|---|---|
Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Harmonic functions, Optimal stopping (Mathematical statistics), Stochastic processes | ||||
Official Date: | April 2020 | ||||
Dates: |
|
||||
Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Assing, Sigurd, 1965- | ||||
Sponsors: | Engineering and Physical Sciences Research Council | ||||
Format of File: | |||||
Extent: | v, 144 leaves : colour illustrations | ||||
Language: | eng |
Request changes or add full text files to a record
Repository staff actions (login required)
View Item |
Downloads
Downloads per month over past year