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Doctoral Thesis
DOI
https://doi.org/10.11606/T.104.2020.tde-21082020-090558
Document
Author
Full name
Danila Maria Silva Fernandes de Almeida
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2020
Supervisor
Committee
Pinto Junior, Dorival Leão (President)
Ferreira, Ricardo Felipe
Ohashi, Alberto Masayoshi Faria
Olivera, Christian Horacio
Ruffino, Paulo Régis Caron
Title in Portuguese
Modelos de Lévy de atividade infinita
Keywords in Portuguese
Equações diferencias estocásticas
Fórmula de Itô
Martingale
Parada ótima
Processos de Lévy
Abstract in Portuguese
Neste trabalho, apresentamos uma classe de processos de Lévy A de puro salto, com filtração interna e decomposição de Itô-Lévy e estabelecemos formas explícitas para a representação martingale, principal componente do nosso processo. Além disso, propomos uma fórmula de Itô-Meyer ótima para um funcional de Lévy e um esquema de aproximação do tipo Euler-Maruyama para uma EDE path-dependent regida pelo processo de Lévy A. Para isso, primeiramente, aproximamos A por um processo de Poisson composto Aε , que provamos convergir fortemente em B2 para A, quando ε ↓ 0. Esse resultado é fundamental para mostrar que, dado um supermartingale envelope de Snell S, podemos aproximá-lo por meio de uma estrutura discreta de encaixe, que vem a ser a sequência de processos valor, associados a S.
Title in English
Infinity Lévy Activity Lévy Models
Keywords in English
Itô formula
Lévy processes
Martingale
Optimal stopping
Stochastic differential equation
Abstract in English
In this work, we present a class of pure jump Lévy processes A, with internal filtration and Itô-Lévy decomposition and we established an explicit forms for martingale representation, main component of our process. Furthermore, we propose an optimal Itô-Meyer formula for a Lévy functional and Euler-Maruyama approach scheme for a path-dependent SDE driven by A Lévy process. For that, first, we close A by a Poisson process composed of Aε, that we proved to converge strongly in B2 to A, when ε ↓ 0. This result is fundamental to show that, given a supermartingale Snell envelope S, we can approach it through an imbedded discrete structure , which is the sequence of value processes, associated with S.
 
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Publishing Date
2020-08-21
 
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