RESOLUTION OF LINEAR EQUATIONS

Author: Rodríguez Nogueiras María.
Year: 2004.
University: SANTIAGO DE COMPOSTELA [www.usc.es].
Place of defense: Facultade de Matemáticas.
Place of preparation: Facultade de Matemáticas.

Summary: This thesis explores the resolution numerical linear equations, and non-linear type convección-difusión-reacción using a method Lagrange-Galerkin of order two. The study was motivated by an application in finance: the valuation of derivative financial products type using equations in partial derivatives. More detail, the method of Lagrange-Galerkin is introduced using the formalism of the mechanics of the continuum and weak formulations. Later, introducing rigorous properties of stability and consistency (order two) of the algorithm. Moreover, it addresses the problem of squaring digital quadrature proposing formulas for finite element Lagrange, linear and quadratic, which preserve the stability of the method (as gauged by a Fourier analysis). Finally, some examples obtained by a numerical code implemented in FORTRAN, illustrating and complete our analysis. With regard to the implementation in finance, is an introduction to modeling in finance within Black-Scholes, focusing on a particular type of financial options: Asian options, which could offset some properties and original formulation mathematical problem of valuing them. In the case of options Asian European style, is the numerical resolution of the valuation problem using methods Lagrange-Galerkin analyzed, taking into account the particularities of specific problem (for example, it optimizes the algebraic structure of the algorithm). In the case of options Asian American type, resolution numerical algorithm combines with iterative methods Lagrange-Galerkin. More precisely, has been compared Two algorithms based on the mixed formulation of the problem: the algorithm Bermúdez-Moreno and an algorithm overall asset type quasi-Newton. In the two cases of Asian options, the numerical results obtained are very satisfactory compared with results obtained using methods Lagrange-Galerkin classics and comparative literature with results obtained with different methods.