Stochastic Processes and Applications

Code:

M4064

Acronym:

M4064

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2021/2022 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Master in Mathematical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:A_ASTR	0	Plano de Estudos oficial desde_2013/14	1	-	6	56	162
			2
M:CC	1	Study plan since 2014/2015	1	-	6	56	162
M:ENM	20	Official Study Plan since 2013-2014	1	-	6	56	162
M:M	3	Plano Oficial do ano letivo 2021	2	-	6	56	162

Teaching language

Suitable for English-speaking students

Objectives

Students should be able to distinguish and recognise the properties of the different stochastic processes studied, such as: the Poisson process, renewal processes, Markov chains and Brownian motion. Students should also be able to use the most common processes as modelling tools. Students should also manage to simulate the different processes and use their respective properties to answer concrete problems.

Learning outcomes and competences

The program includes several tools for the analysis of stochastic processes and its applications in several areas. Special attention is given to the understanding of concepts and methods conducing to its application in interdisciplinary areas. Each method is introduced with examples that are solved in class so that the student has a good understanding of the examples and of their solution.  A parallel supplementary exercise list is also proposed. In addition, the student must develop computational projects where the methodologies introduced are applied to the real world.

Working method

Presencial

Program

Stochastic processes definition. Stationarity. Dependence structures. Examples of stochastic processes used as usual models. Poisson processes: homogeneous and inhomogeneous; transformations of the Poisson process, inter-arrival times distribution, the inspection paradox, the order statistics property, higher dimensional Poisson processes and simulation. Markov chains. Recurrence and transience. Limit Theorem for Markov chains and the Perron-Frobenius Theorem. Applicatiuons to search engines thermodynamics. Brownian motion, properties and simulation. Fractional Brownian motion and applications to the construction of fractal landscapes. Introduction to stichastic calculus. The Itô integral. Stochastic differential equations and applications to finance.

Mandatory literature

Samuel Karlin; A first course in stochastic processes. ISBN: 0-12-398552-8
Howard M. Taylor; An introduction to stochastic modeling. ISBN: 0-12-684885-8
Thomas Mikosch; Elementary stochastic calculus. ISBN: 9810235437

Complementary Bibliography

Thomas Mikosch; Non-life insurance mathematics. ISBN: 9783540882329
J. L. Doob; Stochastic processes. ISBN: 0-471-21813-8
Patrick Billingsley; Probability and measure. ISBN: 0-471-00710-2

Teaching methods and learning activities

Theoretical lectures will be essentially expository with the main purpose of teaching the theoretical background that supports the properties and main results. TP lectures will be used to present and illustrate the main topics by studying examples and solving exercises.

Software

Matlab
(R)

keywords

Physical sciences > Mathematics > Applied mathematics
Physical sciences > Mathematics > Probability theory
Physical sciences > Mathematics > Statistics

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Trabalho escrito	15,00
Exame	40,00
Trabalho prático ou de projeto	45,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	100,00
Frequência das aulas	56,00
Trabalho escrito	3,00
Trabalho laboratorial	3,00
Total:	162,00

Eligibility for exams

Practical assignments/ Project submitted within the fixed deadlines.

Calculation formula of final grade

Distributed evaluation with final exam. Written exam ( E ) and laboratorial work/project (P). Final classification (E*8+P*12)/20. Minimum mark of E and P: 40%.