Probability and Stochastic Processes

Code:

M509

Acronym:

M509

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2015/2016 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Doctoral Program in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
IUD-M	4	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

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.

Learning outcomes and competences

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.

Working method

Presencial

Program

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

Mandatory literature

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

Complementary Bibliography

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

Teaching methods and learning activities

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.

keywords

Physical sciences > Mathematics > Probability theory

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	30,00
Teste	70,00
Total:	100,00

Calculation formula of final grade

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