本文旨在討論非線性拋物型積分微分方程式(組)的解之存在性．首先藉由與上解及下解相關的若干假設，我們得到一個比較性的結果．然後我們利用單調法建構出兩個單調收歛到方程式解的序列，從而驗證了方程式解的存在性． / In this paper, the existence of the solutions for nonlinear integro-differential equations and systems is discussed. First, by the assumption of weak upper and weak lower solutions for the given problem, we obtain the comparison result. Next, we provide the method of monotony and construct two sequences which converge monotonically to the solution.

In this paper, the monotone method is extended to the initial-boundary value problems of nonlocal PDE system of first order, both quasi-monotone and non-monotone. A comparison principle is established, and a monotone scheme is given.

The evolution of the price of two financial assets may be modeled by correlated geometric Brownian motion with additional, independent, finite activity jumps. Similarly, the evolution of the price of one financial asset may be modeled by a stochastic volatility process and finite activity jumps. The value of a contingent claim, written on assets where the underlying evolves by either of these two-factor processes, is given by the solution of a linear, two-dimensional, parabolic, partial integro-differential equation (PIDE). The focus of this thesis is the development of new, efficient numerical solution approaches for these PIDE's for both linear and non-linear cases. A localization scheme approximates the initial-value problem on an infinite spatial domain by an initial-boundary value problem on a finite spatial domain. Convergence of the localization method is proved using a Green's function approach. An implicit, finite difference method discretizes the PIDE. The theoretical conditions for the stability of the discrete approximation are examined under both maximum and von Neumann analysis. Three linearly convergent, monotone variants of the approach are reviewed for the constant coefficient, two-asset case and reformulated for the non-constant coefficient, stochastic volatility case. Each monotone scheme satisfies the conditions which imply convergence to the viscosity solution of the localized PIDE. A fixed point iteration solves the discrete, algebraic equations at each time step. This iteration avoids solving a dense linear system through the use of a lagged integral evaluation. Dense matrix-vector multiplication is avoided by using an FFT method. By using Green's function analysis, von Neumann analysis and maximum analysis, the fixed point iteration is shown to be rapidly convergent under typical market parameters. Combined with a penalty iteration, the value of options with an American early exercise feature may be computed. The rapid convergence of the iteration is verified in numerical tests using European and American options with vanilla payoffs, and digital, one-touch option payoffs. These tests indicate that the localization method for the PIDE's is effective. Adaptations are developed for degenerate or extreme parameter sets. The three monotone approaches are compared by computational cost and resulting error. For the stochastic volatility case, grid rotation is found to be the preferred approach. Finally, a new algorithm is developed for the solution of option values in the non-linear case of a two-factor option where the jump parameters are known only to within a deterministic range. This case results in a Hamilton-Jacobi-Bellman style PIDE. A monotone discretization is used and a new fixed point, policy iteration developed for time step solution. Analysis proves that the new iteration is globally convergent under a mild time step restriction. Numerical tests demonstrate the overall convergence of the method and investigate the financial implications of uncertain parameters on the option value.

The evolution of the price of two financial assets may be modeled by correlated geometric Brownian motion with additional, independent, finite activity jumps. Similarly, the evolution of the price of one financial asset may be modeled by a stochastic volatility process and finite activity jumps. The value of a contingent claim, written on assets where the underlying evolves by either of these two-factor processes, is given by the solution of a linear, two-dimensional, parabolic, partial integro-differential equation (PIDE). The focus of this thesis is the development of new, efficient numerical solution approaches for these PIDE's for both linear and non-linear cases. A localization scheme approximates the initial-value problem on an infinite spatial domain by an initial-boundary value problem on a finite spatial domain. Convergence of the localization method is proved using a Green's function approach. An implicit, finite difference method discretizes the PIDE. The theoretical conditions for the stability of the discrete approximation are examined under both maximum and von Neumann analysis. Three linearly convergent, monotone variants of the approach are reviewed for the constant coefficient, two-asset case and reformulated for the non-constant coefficient, stochastic volatility case. Each monotone scheme satisfies the conditions which imply convergence to the viscosity solution of the localized PIDE. A fixed point iteration solves the discrete, algebraic equations at each time step. This iteration avoids solving a dense linear system through the use of a lagged integral evaluation. Dense matrix-vector multiplication is avoided by using an FFT method. By using Green's function analysis, von Neumann analysis and maximum analysis, the fixed point iteration is shown to be rapidly convergent under typical market parameters. Combined with a penalty iteration, the value of options with an American early exercise feature may be computed. The rapid convergence of the iteration is verified in numerical tests using European and American options with vanilla payoffs, and digital, one-touch option payoffs. These tests indicate that the localization method for the PIDE's is effective. Adaptations are developed for degenerate or extreme parameter sets. The three monotone approaches are compared by computational cost and resulting error. For the stochastic volatility case, grid rotation is found to be the preferred approach. Finally, a new algorithm is developed for the solution of option values in the non-linear case of a two-factor option where the jump parameters are known only to within a deterministic range. This case results in a Hamilton-Jacobi-Bellman style PIDE. A monotone discretization is used and a new fixed point, policy iteration developed for time step solution. Analysis proves that the new iteration is globally convergent under a mild time step restriction. Numerical tests demonstrate the overall convergence of the method and investigate the financial implications of uncertain parameters on the option value.

Orientador: Marcelo Martins dos Santos / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-23T21:53:25Z (GMT). No. of bitstreams: 1 Santos_RonaldoAntoniodos_D.pdf: 739129 bytes, checksum: f677894f21ef1223fced35636869835d (MD5) Previous issue date: 2013 / Resumo: Neste trabalho provamos a existência de solução global para um sistema não linear constituído de duas equações parabólicas acopladas a duas equações diferenciais ordinárias. Tal sistema modela um processo de combustão em um meio poroso com duas camadas, em que os efeitos de compressibilidade são desprezados, mas a troca de calor entre as camadas, bem como a propagação de calor por convecção são levadas em conta. Supondo que os dados iniciais são lipschitzianos, limitados e pertencentes a algum espaço , 1 < < ?, obtivemos solução clássica para o problema / Abstract: In this work we prove the existence of a global solution for a nonlinear system consisting of two parabolic equations coupled to two ordinary differential equations. Such a system models a combustion process in a porous medium with two layers in which compressibility effects are neglected, but heat transfer between the layers as well as heat conduction are taken into a account. We obtained a classical solution under the assumptions that the initial data is bounded, Lipschitz and belongs to some space, with 1 < < ? / Doutorado / Matematica / Doutor em Matemática