Simulation and Stochastic Processes

Code:

M3014

Acronym:

M3014

Level:

300

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2020/2021 - 1S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	6	Official Study Plan	3	-	6	56	162
L:CC	2	Plano de estudos a partir de 2014	2	-	6	56	162
			3
L:EG	1	The study plan from 2019	3	-	6	56	162
L:F	5	Official Study Plan	2	-	6	56	162
			3
L:G	0	study plan from 2017/18	2	-	6	56	162
			3
L:M	43	Official Study Plan	2	-	6	56	162
			3
L:Q	0	study plan from 2016/17	3	-	6	56	162

Teaching language

Portuguese

Objectives

The main objective of the course is to introduce rigorously the main concepts of Stochastic Processes and Simulation. Those concepts and the relevant mathematical tools to their analysis in several applications will be considered in the course.

Learning outcomes and competences

In the first part of the course, some eseential concepts about Monte Carlo methods and Stochastic Processes will be consolidated. The second part of the course will be devoted to applications of the aquired knowledge using simulation in other fields of knowledge.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

It is advised that the student had previous contact with: Probabilities and Statistics, and Real Analysis.

Program

I. Revisions on probabilities and discrete and continuous random variables.

II. Simulation and the Monte Carlo Method Statistical aspects of simulation. Simulation of data (discrete and continuous distributions): general methods, transformations and mixtures; critical use of available current generators. Monte Carlo integration and estimation of expected values. Variance reduction techniques. Monte Carlo method in statistical inference. Resampling methods.

III. Random walk. Browninan motion.

IV. Introduction to stochastic processes and its simulation. Classes of stochastic processes. Introduction to statistical analysis of signals and time series: characterization, stationarity, autocorrelation. 

IV. Estimation and simulation. Modeling/simulation: Markov chains, Poisson process, random walk, birth and death processes, queuing theory.

Mandatory literature

Ross Sheldon M.; Simulation. ISBN: 0-12-598410-3
Papoulis Athanasios; Probability, random variables, and stochastic processes. ISBN: 0-07-048468-6
Shonkwiler Ronald W. 1942-; Explorations in Monte Carlo methods. ISBN: 9780387878362
Law A., Kelton W.D; Simulation Modelling and Analysis, McGrawHill, 2007. ISBN: 978-0073401324
Wood Matt A.; Python and Matplotlib essentials for scientists and engineers. ISBN: 978-1-62705-619-9
Evans Lawrence C. 1949-; An introduction to stochastic differential equations. ISBN: 978-1-4704-1054-4

Complementary Bibliography

Ross Sheldon M.; Introduction to probability models. ISBN: 978-0-12-375686-2

Teaching methods and learning activities

Presentation of the topics of the course and their discussion with the students.

Software

Jupyter
Python

keywords

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

Evaluation Type

Distributed evaluation without final exam

Assessment Components

designation	Weight (%)
Teste	85,00
Trabalho prático ou de projeto	15,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	106,00
Frequência das aulas	56,00
Total:	162,00

Eligibility for exams

Score greater than 9,5 points.

Calculation formula of final grade

Final classification = t1 + t2 + tc1 + tc2
t1 = 1st test score quoted to 8,5
t2 = 2st test score quoted to 8,5
tc1 = 1st computational work quoted to 1,5
tc1 = 2st computational work quoted to 1,5

NOTE: tc1 and tc2 are obtained during class time.
These scores transit to the second season exam.

SECOND SEASON EXAM:
Final classification = er1 + er2 + tc1 + tc2
er1 = 1st exam score quoted to 8,5
er2 = 1st exam score quoted to 8,5
tc1, tc2 = tc1 and tc2 are obtained during class time.

(1) The second season exam consists of two parts corresponding to the division of matter for the tests.

(2) In the second season exam, the student can choose one or two of its parts. If he/she submits it for correction, it will replace the corresponding classification(s) obtained in the test(s).

Note: t1 and/or t2 can be divided into two parts

Classification improvement

Grade improvement will be made in the appeal examination.