Disciplina Discipline MAE5815
Probabilidade Avançada II

Advanced Probability II

Área de Concentração: 45133

Concentration area: 45133

Criação: 24/07/2015

Creation: 24/07/2015

Ativação: 24/07/2015

Activation: 24/07/2015

Nr. de Créditos: 8

Credits: 8

Carga Horária:

Workload:

Teórica

(por semana)

Theory

(weekly)

Prática

(por semana)

Practice

(weekly)

Estudos

(por semana)

Study

(weekly)

Duração Duration Total Total
4 2 4 12 semanas 12 weeks 120 horas 120 hours

Docentes Responsáveis:

Professors:

Luiz Renato Goncalves Fontes

Vladimir Belitsky

Objetivos:

Complementar as noções básicas da Teoria das Probabilidades adquiridas no curso de Probabilidade Avançada I, introduzindo tópicos especiais da teoria moderna de probabilidade, centradas nas noções de Martingais, Teoria Ergódica e elementos de Cálculo Estocástico.

Objectives:

Complement the basics of Probability Theory acquired in the course of Advanced Probability I, introducing special topics of modern probability theory, centered on the notions of Martingals, Ergodic Theory and elements of Stochastic Calculus.

Justificativa:

Os tópicos tratados neste cursos constituem ferramentas indispensáveis para o trabalho de pesquisa em Teoria de Probabilidades e Teoria Estatística, assim como para aplicações modernas em áreas tão variadas como tratamento do sinal, finanças, etc

Rationale:

The topics covered in this course are indispensable tools for the research work in Probability Theory and Statistical Theory, as well as for modern applications in areas as varied as signal processing, finance, etc.

Conteúdo:

1. Martingais (a) Convergência Quase-Certa (b) Desigualdade de Doob, Convergência em Lp (c) Integrabilidade Uniforme, Convergência em L1 (d) Teorema da Parada Ótima 2. Processos Estacionários e Teorema Ergódico de Birkhoff 3. Movimento Browniano (a) Construção (b) Propriedade de Markov, Princípio da Reflexão (c) Tempos de Passagem (d) Propriedades das Trajetórias 4. Integração Estocástica (a) Construção da Integral Estocástica (b) Fórmula de Itô, Teorema da Girsanov

Content:

1. Martingales (a) Almost-Sure Convergence (b) Doob's Inequality, Convergence in Lp (c) Uniform Integrability, Convergence in L1 (d) Optimal Stopping Theorem 2. Stationary Processes and Birkhoff's Ergodic Theorem 3. Brownian Motion (a) Construction (b) Markov Property, Reflection Principle (c) Passage Times (d) Properties of the Trajectories 4. Stochastic Integration (a) Construction of the Stochastic Integral (b) Itô's Formula, Girsanov's Theorem

Forma de Avaliação:

A avaliação consistirá da média de provas e listas de exercícios.

Type of Assessment:

Exams and Worksheets

Observação:

Bibliografia:

1. Durret, R. (1996). Probability: Theory and Examples. Second Edition, Duxbury Press. 2. Shiryaev, A. N. (1996). Probability. Second edition. Springer. 3. Chung, K.L. (1974). A Course in Probability Theory. Second Edition, Academic Press. 4. Breiman, L. (1968). Probability. Addison-Wesley (republicado por SIAM). 5. Billingsley, P. (1995). Probability and Measure. Third edition. Wiley. 6. Feller, W. (1971). An Introduction to Probability Theory and its Applications. Vol. I e Vol. II, Second Edition, Wiley. 7. Lamperti, J. (1966). Probability: A Survey of the Mathematical Theory. Benjamin. 8. Oksendal, B. K. (1998) Stochastic Differential Equations: An Introduction With Applications Springer. 9. Karatzas, I.; Shreve, S.E(1988). Brownian Motion and Stochastic Calculus. Springer. 10. Durrett, R.; Pinsky, M.(1996) Stochastic Calculus: A Practical Introduction. CRC Press

Bibliography:

1. Durret, R. (1996). Probability: Theory and Examples. Second Edition, Duxbury Press. 2. Shiryaev, A. N. (1996). Probability. Second edition. Springer. 3. Chung, K.L. (1974). A Course in Probability Theory. Second Edition, Academic Press. 4. Breiman, L. (1968). Probability. Addison-Wesley (republicado por SIAM). 5. Billingsley, P. (1995). Probability and Measure. Third edition. Wiley. 6. Feller, W. (1971). An Introduction to Probability Theory and its Applications. Vol. I e Vol. II, Second Edition, Wiley. 7. Lamperti, J. (1966). Probability: A Survey of the Mathematical Theory. Benjamin. 8. Oksendal, B. K. (1998) Stochastic Differential Equations: An Introduction With Applications Springer. 9. Karatzas, I.; Shreve, S.E(1988). Brownian Motion and Stochastic Calculus. Springer. 10. Durrett, R.; Pinsky, M.(1996) Stochastic Calculus: A Practical Introduction. CRC Press