Probability and Stochastic Processes

Code:

M509

Acronym:

M509

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2020/2021 - 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	6	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, Skorokhod'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 of advanced topics in Probability Theory, which includes getting acquainted with certain tools such as Kolmogorov's existence theorem, Skorokhod'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 Skorokhod's representation
2.2 Kolmogorov's existence theorem
2.3 Independence
2.4 Kolmogorv's 0-1 Law
2.5 Borel-Cantelli Lemmas
2.6 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

Lectures  where the topics of the syllabus are presented, exercises and related problems are solved and discussed. The worked exercises, examples and problems are fundamental to help the understanding of the concepts and to illustrate their potential of application. Project work must be done by each student, involving a written report and posterior oral presentation and discussion.

keywords

Physical sciences > Mathematics > Probability theory

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	60,00
Trabalho escrito	40,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Apresentação/discussão de um trabalho científico	4,00
Estudo autónomo	183,00
Frequência das aulas	56,00
Total:	243,00

Eligibility for exams

Practical assignments/ Project submitted within the fixed  schedules.

Calculation formula of final grade

The final mark is obtained by an weighted average of the written assignment (40%) and the final exam (60%). The minimum score in each component is 50% of the corresponding value