Probability and Stochastic Processes

Recommended Prerequisites

The student should be acquainted with Real Analysis and Lebesgue Integration.

Teaching Methods

Lectures have an expository character, being up to the professor the choice of the most appropriate way to do it and the degree of participation of the students. As an integral part of the learning process, it may be recommended or required 

the solution of exercises, course projects (possibly with a computational component) or oral presentations.

Learning Outcomes

The goal of this course is to provide a general knowledge of Probability and Stochastic Processes at an advanced level. It is intended that the students become familiar with the main basic techniques and results of this area of Mathematics, or at least attain enough familiarity with some of  them to be able to acquire others by themselves that may later prove to be useful.

Work Placement(s)

No

Syllabus

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 Borel-Cantelli Lemmas

2.5 Kolmogorv's 0-1 Law

2.4 Conditional expectation

3 Martingales

3.1 Definitions and properties

3.2 Stopping times and inequalities

3.3 (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

Head Lecturer(s)

Paulo Eduardo Aragão Aleixo e Neves de Oliveira

Assessment Methods

Assessment 2
Frequency: 50.0%
Exam: 50.0%

Assessment 1
Frequency: 40.0%
Exam: 60.0%

Bibliography

P. Billingsley, Probability and Measure, 3rd edition, John Wiley & Sons Inc., New York, 1995.

P. Billingsley, Convergence of Probability Measures, 2nd edition, John Wiley & Sons Inc., New York, 1999.

J.F.C. Kingman and S.J. Taylor, Introduction to Measure and Probability, Cambridge University Press, London, 1966.

D.W. Stroock, Probability theory, an analytic view, Cambridge University Press, Cambridge, 1993.

S.R.S. Varadhan, Probability theory, Courant Lecture Notes in Mathematics, vol. 7, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001.

S.R.S. Varadhan. Stochastic Processes, Courant Lecture Notes in Mathematics, vol. 16, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2007.