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Convergence analysis of monotone schemes for second-order non-linear parabolic PDEs and their applications in sublinear expectation
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Huang, Shuo (2020) Convergence analysis of monotone schemes for second-order non-linear parabolic PDEs and their applications in sublinear expectation. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3520397~S15
Abstract
Second-order non-linear parabolic partial differential equations have been a central research area for decades due to their various applications to physics, engineering, finance and others. To solve such equations, numerical approximations are usually designed to converge to a weak version of solutions called viscosity solutions. This thesis proposes monotone approximation schemes for two specific types of non-linear parabolic equations arising in applied mathematics, and establishes the convergence and convergence rate to viscosity solutions. The proposed schemes involve only semi-discretization in time and the rates of convergence are represented in the form of some exponent of the time step scaled by a constant.
We first propose a monotone scheme for a class of semi-linear parabolic equations that are convex and coercive in their gradients arising from utility indifference pricing in mathematical finance. The proposed scheme is based on a splitting method and its convergence rate is determined by combining Krylov’s shaking coefficients technique and the Barles-Jakobsen optimal switching approximation. An extension to variational inequalities is also studied using an obstacle switching system. We then build a piece-wise constant monotone scheme for a class of fully non-linear equations called G-equations arising from Knightian uncertainty in statistics, and determine its convergence rate with an explicit error bound. We present three applications under a sublinear expectation framework: a convergence rate for Peng’s robust central limit theorem with an explicit bound of Berry-Esseen type, a monoton scheme for the Black-Scholes-Barenblatt (BSB) equation that is a natural generalization of Cox-Ross-Rubinstein (CIR) binomial tree approximation to the case with model ambiguity, and an optimal switching approximation to G-normal distribution.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Differential equations, Parabolic, Differential equations, Nonlinear, Monotone operators, Viscosity solutions | ||||
Official Date: | October 2020 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Department of Statistics | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Liang, Gechun | ||||
Sponsors: | University of Warwick. Department of Statistics | ||||
Format of File: | |||||
Extent: | vi, 123 leaves : colour illustrations | ||||
Language: | eng |
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