Sun Siu-wing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 50-51). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Equation without viscosity --- p.5 / Chapter 3 --- Equation with standard viscosity --- p.8 / Chapter 3.1 --- "Particular convective flux, f(x) =u2" --- p.8 / Chapter 3.2 --- Convex convective flux --- p.10 / Chapter 4 --- Equation with monotonic dissipative flux --- p.11 / Chapter 4.1 --- Large initial data --- p.12 / Chapter 4.2 --- Small initial data --- p.19 / Chapter 4.3 --- Unbounded dissipative flux --- p.28 / Chapter 5 --- Equation with non-monotonic dissipative flux --- p.31 / Chapter 5.1 --- Large initial data --- p.32 / Chapter 5.2 --- Small initial data --- p.37 / Chapter 6 --- Comparison and conclusions --- p.39 / Appendices --- p.42 / Chapter A --- Hopf-Cole transformation --- p.42 / Chapter B --- Dirichlet problem --- p.45 / Bibliography --- p.50

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2001. / Includes bibliographical references. Also available on the Internet.

In this thesis, we establish the asymptotic analysis of the singularly perturbed reaction diffusion equation [cataloger unable to transcribe mathematical equations].... Our results establish the specific dependency on the coefficients of this equation and the size of the parameter [delta] with respect to [epsilon]. The analyses include equation subject to Dirichlet and Neumann boundary conditions. In both cases, the solutions u[superscript epsilon] converge locally uniformally to the equilibria of the reaction term f. We characterize the limiting behavior of the solutions through the viscosity solution of a variational inequality. To construct the coefficients defining the variational inequality, we apply concepts developed for the homogenization of elliptic operators. In chapter two, we derive the convergence results in the Neumann case. The third chapter is dedicated to the analysis of the Dirichlet case. / text

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

published_or_final_version / Physics / Doctoral / Doctor of Philosophy

Thesis (Ph. D.)--University of Hong Kong, 1998. / Includes bibliographical references.

Thesis (Ph. D.)--University of Chicago, Dept. of Economics. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph.D.)--Brown University, 2005. / Vita. Thesis advisor: Chi-Wang Shu. Includes bibliographical references (leaves 83-87). Also available online.

In this thesis, we consider second-order fully nonlinear partial differential equations of elliptic type. Our aim is to develop computational methods using convergent difference schemes for stationary Hamilton-Jacobi equations with Dirichlet and Neumann type boundary conditions in arbitrary two-dimensional domains. First, we introduce the notion of viscosity solutions in both continuous and discontinuous frameworks. Next, we review Barles-Souganidis approach using monotone, consistent, and stable schemes. In particular, we show that these schemes converge locally uniformly to the unique viscosity solution of the first-order Hamilton-Jacobi equations under mild assumptions. To solve the scheme numerically, we use Euler map with some initial guess. This iterative method gives the viscosity solution as a limit. Moreover, we illustrate our numerical approach in several two-dimensional examples.

This thesis explores numerical methods for solving nonlinear partial differential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and efficient techniques that converge to the financially relevant solution for both one and two factor problems. To illustrate the underlying concepts, two nonlinear models are examined in detail: uncertain volatility and passport options. For any nonlinear model, implicit timestepping techniques lead to a set of discrete nonlinear equations which must be solved at each timestep. Several iterative methods for solving these equations are tested. In the cases of uncertain volatility and passport options, it is shown that the frozen coefficient method outperforms two different Newton-type methods. Further, it is proven that the frozen coefficient method is guaranteed to converge for a wide class of one factor problems. A major issue when solving nonlinear PDEs is the possibility of multiple solutions. In a financial context, convergence to the viscosity solution is desired. Conditions under which the one factor uncertain volatility equations are guaranteed to converge to the viscosity solution are derived. Unfortunately, the techniques used do not apply to passport options, primarily because a positive coefficient discretization is shown to not always be achievable. For both uncertain volatility and passport options, much work has already been done for one factor problems. In this thesis, extensions are made for two factor problems. The importance of treating derivative estimates consistently between the discretization and an optimization procedure is discussed. For option pricing problems in general, non-smooth data can cause convergence difficulties for classical timestepping techniques. In particular, quadratic convergence may not be achieved. Techniques for restoring quadratic convergence for linear problems are examined. Via numerical examples, these techniques are also shown to improve the stability of the nonlinear uncertain volatility and passport option problems. Finally, two applications are briefly explored. The first application involves static hedging to reduce the bid-ask spread implied by uncertain volatility pricing. While static hedging has been carried out previously for one factor models, examples for two factor models are provided. The second application uses passport option theory to examine trader compensation strategies. By changing the payoff, it is shown how the expected distribution of trading account balances can be modified to reflect trader or bank preferences.