Probabilidades e Processos Estocásticos

Código:

M509

Sigla:

M509

Áreas Científicas
Classificação	Área Científica
OFICIAL	Matemática

Ocorrência: 2015/2016 - 1S

Ativa?	Sim
Unidade Responsável:	Departamento de Matemática
Curso/CE Responsável:	Programa Doutoral em Matemática - Interuniversitário

Ciclos de Estudo/Cursos

Sigla	Nº de Estudantes	Plano de Estudos	Anos Curriculares	Créditos UCN	Créditos ECTS	Horas de Contacto	Horas Totais
IUD-M	4	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Língua de trabalho

Inglês

Objetivos

The main goal of the course is to give the foundations of modern Probability Theory.

The first objective is to make a brief introduction to measure theory and integration meant to recall concepts and uniforming the students background.

The course is designed to guarantee that the students learn important tools and concepts used frequently in Probability Theory and its applications. Namely: Kolmogorov’s 0-1 law, Skhorokhod's representation and embedding, tightness and Prokhorov's theorem, invariance principles and Donsker’s theorem, just to mention a few.

Moreover, another important goal is the study of special processes such as Martingales and Brownian motion, their properties and range of applications.

Resultados de aprendizagem e competências

The student should acquire knowledge ofadvanced topics in Probability Theory, which includes getting acquianted with certain tools such as Kolmogorov's existence theorem, Skhorokhod's representation and embedding, tightness and Prokhorov's theorem, invariance principles and Donsker’s theorem.

Moreover the student should learn about special processes such as Martingales and Brownian motion and their powerful spectrum of applications.

Modo de trabalho

Presencial

Programa

1  Preliminaries
1.1 Probability spaces
1.2 Integration
1.3 Absolute continuity
1.4 Notions of convergence and Slutsky’s theorem

2 Random variables and Stochastic processes
2.1 Distributions and Skhorokhod's representation
2.2 Kolmogorov's existence theorem
2.3 Independence
2.4 Kolmogorv's 0-1 Law
2.5 Borel-Cantelli Lemmas
2.4 Conditional expectation

3 Martingales
3.1 Definitions and properties
3.2 Stopping times and inequalities
3.2 (Sub)martingale convergence theorem
3.4 Central limit theorem
3.5* Application to mixing stationary processes (the Gordin approximation)

4 Brownian motion
4.1 Continuity of paths and their irregularity
4.2 Strong Markov property and reflection principle
4.3 Skorohod's Embedding

5 Weak convergence
5.1 Portmanteau theorem
5.2 Tightness and Prokhorov's theorem
5.3 Weak convergence in C[0,1]
5.4 Donsker's theorem and Invariance principle

Bibliografia Obrigatória

Billingsley Patrick; Probability and measure. ISBN: 0-471-00710-2

Bibliografia Complementar

Billingsley Patrick; Convergence of probability measures
Kallenberg Olav; Foundations of modern probability. ISBN: 978-1-4419-2949-5
Kingman J. F. C. (John Frank Charles); Introduction to measure and probability. ISBN: 0-521-05888-0
S. R. S. Varadhan; Probability theory, 2001. ISBN: 0-8218-2852-5
S. R. S. Varadhan; Stochastic processes, 2007. ISBN: 978-0-8218-4085-6
D. W. Stroock; Probability theory, 1993. ISBN: 0-521-43123-9

Métodos de ensino e atividades de aprendizagem

Each class will last for two hours. Whenever possible, the last half hour of each class will be used to solve examples / exercises in order to help the understanding of the concepts studied in the first part of the lesson, to illustrate the potential of application, and also also  stimulate interest in discipline. Exercise sheets will be made available.

Palavras Chave

Ciências Físicas > Matemática > Teoria das probabilidades

Tipo de avaliação

Avaliação distribuída com exame final

Componentes de Avaliação

Designação	Peso (%)
Exame	30,00
Teste	70,00
Total:	100,00

Fórmula de cálculo da classificação final

The mark obtained in the final exam will determin the student's classification.