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Master's Dissertation
DOI
10.11606/D.45.2009.tde-07122009-131027
Document
Author
Full name
Fabio Niski
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2009
Supervisor
Committee
Faria, Edson de (President)
Alfonso, Nestor Felipe Caticha
Dreifus, Henrique Von
Title in Portuguese
Integral estocástica e aplicações
Keywords in Portuguese
Cálculo Estocástico
Fórmula de Feynman Kac
Fórmula de Itô
Integral Estocástica
Movimento Browniano
Abstract in Portuguese
O aumento pelo interesse na teoria de integração estocástica é, basicamente, consequência da acirrada competição para entender, desenvolver e aplicar a matemática subjacente ao mercado mobiliário. Neste trabalho desenvolvemos, de maneira didática e visando aplicações, tal teoria. Para tanto, começamos apresentando um desenvolvimento cuidadoso da teoria dos martingais e dos principais resultados de medida e probabilidade relacionados. Depois apresentamos de maneira formal a teoria de integração estocástica com respeito aos semi-martingais contínuos. Finalizamos com um tratamento das principais aplicações dessa teoria como a fórmula de Itô, uma introdução às equações diferenciais estocásticas e a fórmula de Feynman-Kac. Apresentamos também, em um apêndice, a teoria de mudança de medida e o teorema de Girsanov. Tentamos durante o trabalho apresentar exemplos relacionados com finanças e ilustrar a importância do movimento Browniano.
Title in English
Stochastic Integral and Applications
Keywords in English
Brownian motion
Feynman-Kac's Formula
Finance
Itô's Formula
Stochastic Integral
Abstract in English
The increasing interest in the theory of Stochastic Integration is due mainly to the competitive pressure to understand, develop and apply the underlying mathematics of security markets. In this work, we attempt to develop part of the theory in a didactical approach and focused toward some particular applications. For this purpose, we begin by introducing a thorough development of Martingale theory and the main related results on Measure and Probability theory. We then present in a formal way the Stochastic Integration Theory with respect to continuous Semimartingales. Subsequentially, we show some of the theory's main applications, such as Itô's formula, an introduction to the theory of Stochastic Differential Equations and Feynman-Kac's formula. We also present in the appendix Girsanov's theorem and a construction of Brownian motion. During the development of this text we endeavored to enrich it by including examples relevant to finance and emphasizing the importance of the ubiquitous Brownian motion.
 
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calculo_estoc.pdf (1.36 Mbytes)
Publishing Date
2010-09-21
 
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