In 1948, Richard Feynman published the first paper on his new approach to non-relativistic quantum mechanics. Before Feynman's work there were two mathematical formulations of quantum mechanics. Schrödinger's formulation was based on PDE (the Schrödinger equation) and states representation by wave functions, so it was in the framework of analysis and differential equations. The other formulation was Heisenberg's matrix algebra. Initially, they were thought to be competing. The proponents of one claimed that the other was “ wrong. ” Within a couple of years, John von Neumann had proved that they are equivalent. Although Feynman's theory was not fundamentally new, it nonetheless offered an entirely fresh and different perspective: via a precise formulation of Bohr's correspondence principle, it made quantum mechanics similar to classical mechanics in a precise sense. In addition, Feynman's approach made it possible to explain physical experiments, and, via diagrams, link them directly to computations. What resulted was a very powerful device for computing energies and scattering amplitudes - the famous Feynman's diagrams. In his formulation, Feynman aimed at representing the solution to the non-relativistic Schrödinger equation in the form of an “ average ” over histories or paths of a particle. This solution is commonly known as the Feynman path integral. It plays an important role in the theory but appears as a postulate based on intuition coming from physics rather than a justified mathematical object. This is why Feynman's vision has caught the attention of many mathematicians as well as physicists. The papers of Gelfand, Cameron, and Nelson are among the first, and more substantial, attempts to supply Feynman's theory with a rigorous mathematical foundation. These attempts were followed by many others, but unfortunately all of them were not quite satisfactory. The difficulty comes from a need to define a measure on an infinite-dimensional space of continuous functions that represent all possible paths of a particle. This Feynman's measure has to produce an integral with the properties requested by Feynman. In particular, the expression for the Feynman measure has to involve the non-absolutely integrable Fresnel integrands. The non-absolute integrability of the Fresnel integrands makes the measure fail to be positive and to have the countably additive property. Thus, a well-defined measure in the case of the Feynman path integral does not exist. Extensive research has been done on the methods of relating the Feynman path integral to the integral with respect to the Wiener measure. The method of analytic continuation in mass defines the Feynman path integral as a certain limit of the Wiener integrals. Unfortunately, this method can be used as definition for only almost all values of the mass parameter in the Schrödinger equation. For physicists, this is not a satisfactory result and needs to be improved. In this work we examine those questions which originally led to the Feynman path integral. By now we know that Feynman's “ dream ” cannot be realized as a positive and countably additive measure on the path-space. Here, we offer a new way out by modifying Feynman's question, and thereby achieving a solution to the Schrödinger equation via a different kind of averages in the path-space. We give our version of the question that Feynman “ should have asked ” in order to realize the elusive path integral. In our formulation, we get a Feynman path integral as a limit of linear functionals, as opposed to the more familiar inductive limits of positive measures, traditionally used for constructing the Wiener measure, and related Gaussian families. We adapt here an approach pioneered by Patrick Muldowney. In it, Muldowney suggested a Henstock integration technique in order to deal with the non-absolute integrability of the kind of Fresnel integrals which we need in our solution to Feynman's question. By applying Henstock's theory to Fresnel integrals, we construct a complex-valued “ probability distribution functions ” on the path-space. Then we use this “ probability ” distribution function to define the Feynman path integral as an inductive limit. This establishes a mathematically rigorous Feynman limit, and at the same time, preserves Feynman's intuitive idea in resulting functional. In addition, our definition, and our solution, do not place any restrictions on any of the parameters in the Schrödinger equation, and have a potential to offer useful computational experiments, and other theoretical insights.

Being part of a research group on functional differential equations (FDEs, for short), due to my formation in non-absolute integration theory and because certain kinds of FDEs can be expressed as integral equations, I was motivated to investigate the latter. The purpose of this work, therefore, is to develop the theory of integral equations, when the integrals involved are in the sense of Kurzweil- Henstock or Kurzweil-Henstock-Stieltjes, through the correspondence between solutions of integral equations and solutions of generalized ordinary differential equations (we write generalized ODEs, for short). In order to be able to obtain results for integral equations, we propose extensions of both the Kurzweil integral and the generalized ODEs (found in [36]). We develop the fundamental properties of this new generalized ODE, such as existence and uniqueness of solutions results, and we propose stability concepts for the solutions of our new class of equations. We, then, apply these results to a class of nonlinear Volterra integral equations of the second kind. Finally, we consider a model of population growth (found in [4]) that can be expressed as an integral equation that belongs to this class of nonlinear Volterra integral equations. / Sendo parte de um grupo de pesquisa em equações diferenciais funcionais (escrevemos EDFs), por causa de minha formação em teoria de integração não absoluta e porque certos tipos de EDFs podem ser escritas como equações integrais, decidi estudar esse último tipo de equações. O objetivo desse trabalho, portanto, é desenvolver a teoria de equações integrais, quando as integrais envolvidas são no sentido de Kurzweil-Henstock ou Kurzweil-Henstock-Stieltjes, através da correspondência entre soluções de equações integrais e soluções de equações diferenciais ordinárias generalizadas (ou EDOs generalizadas). A fim de obter resultados para estas equações integrais, propomos extensões de ambas a integral de Kurzweil e as EDOs generalizadas (encontradas em [36]). Desenvolvemos propriedades fundamentais dessa nova EDO generalizada, como resultados de existência e unicidade de solução, e propomos conceitos de estabilidade para as soluções de nossa nova classe de equações. Nós, então, aplicamos esses resultados a uma classe de equações integrais de Volterra não lineares de segunda espécie. Finalmente, consideramos um modelo de crescimento de populações (encontrado em [4]) que pode ser escrito como uma equação integral pertencente a essa classe de equações integrais de Volterra não lineares.

Impulsos são perturbações abruptas que ocorrem em curto espaço de tempo e podem ser consideradas instantâneas. E os mercados financeiros estão sujeitos a choques bruscos como mudanças de governos, quebra de empresas, entre outros. Assim, é natural considerarmos a ação de tais eventos na precificação de ativos financeiros. Nosso objetivo neste trabalho é obtermos uma formulação para a equação diferencial parcial de Black-Scholes com ação impulsiva de modo que os impulsos representem estes choques. Utilizaremos a teoria de integração não-absoluta em espaço de funções para obtenção desta formulação / Impulses describe the evolution of systems where the continuous development of a process is interrupted by abrupt changes of state. Financial markets are subject to extreme events or shocks as government changes, companies colapse, etc. Thus it seems natural to consider the action of these events in the valuation of derivative securities. The aim of this work is to obtain a formulation for the Black-Scholes equation with impulse action where the impulses can represent these shocks. We use the non-absolute integration theory in functional spaces to obtain such formulation

Being part of a research group on functional differential equations (FDEs, for short), due to my formation in non-absolute integration theory and because certain kinds of FDEs can be expressed as integral equations, I was motivated to investigate the latter. The purpose of this work, therefore, is to develop the theory of integral equations, when the integrals involved are in the sense of Kurzweil- Henstock or Kurzweil-Henstock-Stieltjes, through the correspondence between solutions of integral equations and solutions of generalized ordinary differential equations (we write generalized ODEs, for short). In order to be able to obtain results for integral equations, we propose extensions of both the Kurzweil integral and the generalized ODEs (found in [36]). We develop the fundamental properties of this new generalized ODE, such as existence and uniqueness of solutions results, and we propose stability concepts for the solutions of our new class of equations. We, then, apply these results to a class of nonlinear Volterra integral equations of the second kind. Finally, we consider a model of population growth (found in [4]) that can be expressed as an integral equation that belongs to this class of nonlinear Volterra integral equations. / Sendo parte de um grupo de pesquisa em equações diferenciais funcionais (escrevemos EDFs), por causa de minha formação em teoria de integração não absoluta e porque certos tipos de EDFs podem ser escritas como equações integrais, decidi estudar esse último tipo de equações. O objetivo desse trabalho, portanto, é desenvolver a teoria de equações integrais, quando as integrais envolvidas são no sentido de Kurzweil-Henstock ou Kurzweil-Henstock-Stieltjes, através da correspondência entre soluções de equações integrais e soluções de equações diferenciais ordinárias generalizadas (ou EDOs generalizadas). A fim de obter resultados para estas equações integrais, propomos extensões de ambas a integral de Kurzweil e as EDOs generalizadas (encontradas em [36]). Desenvolvemos propriedades fundamentais dessa nova EDO generalizada, como resultados de existência e unicidade de solução, e propomos conceitos de estabilidade para as soluções de nossa nova classe de equações. Nós, então, aplicamos esses resultados a uma classe de equações integrais de Volterra não lineares de segunda espécie. Finalmente, consideramos um modelo de crescimento de populações (encontrado em [4]) que pode ser escrito como uma equação integral pertencente a essa classe de equações integrais de Volterra não lineares.

Impulsos são perturbações abruptas que ocorrem em curto espaço de tempo e podem ser consideradas instantâneas. E os mercados financeiros estão sujeitos a choques bruscos como mudanças de governos, quebra de empresas, entre outros. Assim, é natural considerarmos a ação de tais eventos na precificação de ativos financeiros. Nosso objetivo neste trabalho é obtermos uma formulação para a equação diferencial parcial de Black-Scholes com ação impulsiva de modo que os impulsos representem estes choques. Utilizaremos a teoria de integração não-absoluta em espaço de funções para obtenção desta formulação / Impulses describe the evolution of systems where the continuous development of a process is interrupted by abrupt changes of state. Financial markets are subject to extreme events or shocks as government changes, companies colapse, etc. Thus it seems natural to consider the action of these events in the valuation of derivative securities. The aim of this work is to obtain a formulation for the Black-Scholes equation with impulse action where the impulses can represent these shocks. We use the non-absolute integration theory in functional spaces to obtain such formulation