A pipeline for high precision partial differential equation computation

Thesis

In this thesis, we investigate the application of the cubature on Wiener space method(also known as the Kusuoka-Lyons-Victoir algorithm) to solving parabolic type lin-ear partial differential equation problems. The framework for the cubature methodwas first developed in the seminal paper [Lyons and Victoir, 2004]. Since then,several key ideas have been introduced to improve the method’s computationaltractability, hence practicality, without affecting its approximation order. We re-view these ideas in chapter 1.

Building on those works, in this thesis, we investigate the use of nonlinear adaptiveapproximation in the cubature framework, to further extend cubature’s practicalapplicability. In particular, we utilise adaptive approximation as an additional com-putation pruning device, so that existing techniques could be successfully applied tocertain Cauchy problems that do not have a predetermined boundary. Example ofsuch problems include compound options and certain path dependent options fromfinance. Using our ideas, we demonstrate using non-trivial numerical test examples(see chapter 3), that we were able to achieve more than two orders of magnitudereduction in overall computation – shortening the “naive” approach, which wouldhave required weeks/months, to minutes on consumer hardware – without affectingthe order of approximation. The amalgamation of cubature and adaptive approx-imation relies on our carefully designed partitioning and memoisation algorithms.Naive treatment of such is well known to suffer from the curse of dimensionality.We explain these ideas in chapter 2.

In summary, this thesis demonstrates the pipeline of carefully designed techniquesthat are necessary for the successful application of the high order cubature onWiener space method to practical problems.

Authors: Pan, W

• DOI: 10.5287/ora-mprwyk8zx
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: high order weak approximation of stochastic differential equations
• Subjects: Stochastic differential equations--Numerical solutions; Differential equations, Partial--Numerical solutions