Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance

Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Bean. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature

Applications of Mathematical Models in Engineering

The most influential research topic in the twenty-first century seems to be mathematics, as it generates innovation in a wide range of research fields. It supports all engineering fields, but also areas such as medicine, healthcare, business, etc. Therefore, the intention of this Special Issue is to deal with mathematical works related to engineering and multidisciplinary problems. Modern developments in theoretical and applied science have widely depended our knowledge of the derivatives and integrals of the fractional order appearing in engineering practices. Therefore, one goal of this Special Issue is to focus on recent achievements and future challenges in the theory and applications of fractional calculus in engineering sciences. The special issue included some original research articles that address significant issues and contribute towards the development of new concepts, methodologies, applications, trends and knowledge in mathematics. Potential topics include, but are not limited to, the following: Fractional mathematical models; Computational methods for the fractional PDEs in engineering; New mathematical approaches, innovations and challenges in biotechnologies and biomedicine; Applied mathematics; Engineering research based on advanced mathematical tools

Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance

Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.South Afric

Nonlinear Analysis and Optimization with Applications

Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world

Stability of the Rayleigh-Ritz-Galerkin procedure for elliptic boundary value problems

This thesis investigates the stability of the Rayleigh-Ritz-Galerkin procedure for the approximate solution of certain classes of linear and nonlinear elliptic boundary value problems. In numerical analysis literature over the last decade, piecewise Hermite and spline subspaces have often been proposed for the Rayleigh-Ritz-Galerkin procedure for the solution of elliptic boundary value problems . However, the use of pie cewise polynomial subspaces has not been investigated f rom the point of view of Mikhlin stability , and this thesis rectifies this neglect in the literature . In Chapter 2, we introduce the Rayleigh-Ritz, the Galerkin, the generalized Rayleigh-Ritz, and the generalized Galerkin methods for the approximate solution of linear operator equations. As well as the concept of Mikhlin stability for linear numerical processes, we also introduce Tucker stability for nonlinear numeric~l processes. Chapter 3 is concerned with certain classes of linear elliptic boundary value problems, where for each class, we establish basic stability theorems and then investigate the Mikhlin stability of the Rayleigh-Ritz-Galerkin procedure when the coordinate functions are appropriately scaled B-splines or elementary Hermites. The three classes that we consider are one dimensional, two dimensional, and multidimensional elliptic boundary value problems with Dirichlet boundary conditions. Chapter 4 is similar to Chapter 3 except that in this case, we are concerned with nonlinear elliptic boundary value problems. The first and second class considered are nonlinear two-point boundary value problems with Dirichlet and nonlinear boundary conditions, respectively. We also study a "model" nonlinear multidimensional problem. In Chapter 5, we study normalized uniformly asymptotically diagonal systems from the point of view of Mikhlin stability, and illustrate the type of instability that can arise with a numerical example