Stochastic Processes


This course intends to give appropriate knowledge foundations for the study of the evolution of random phenomena.

At the end of this course, the student will obtain knowledge, skills and competences that allow him to:

-Recognize and use the main properties of chosen examples of time discrete stochastic processes with special emphasis on applications;

- To be able to decide which is the more appropriate model of a stochastic process to use when faced with a realistic situation.

- Identify the phenomena adequate to be modeled by Poisson Processes and make use of the properties, giving special emphasis to real applications.

- To identify a Markov chain and use the characteristic properties of this type of processes for the analysis of a concrete model. Perform applications to real and concrete problems.

- To identify a martingale and use the characteristic properties of this type of processes in the study of its behavior, in particular, in the determination of a possible asymptotic behavior,

General characterization





Responsible teacher

Luís Pedro Carneiro Ramos


Weekly - 4

Total - 62

Teaching language



The student should have knowledge about Probability Theory.


Hastings, K., Introduction to Probability with Mathematica, 2nd Ed., CRC Presss, Chapman & Hall, 2010

Muller, D, Processos Estocásticos e Aplicações, Edições Almedian, 2007

Norris, J.R., Markov Chains, Cambridge University Press, 1997.

Parzen, E., Stochastic Processes, Holden Day, 1965

Rohatgi, V.K, Saleh, A.K, An Introduction to Probability and Statistics, 2nd Ed, Wiley Series in Probability and Statistics, 2001 (para revisões de Probabilidades e Estatística)

Ross, S. M., Stochastic Processes, 2nd Ed., Wiley & Sons, 1996

Williams, D., Probability with Martingales, Cambridge University Press, 1991.

Teaching method

The main goal of the Curricular Unit is to provide students with tools to model stochastic phenomena.

It is intended that students acquire theoretical and practical skills that allow them to understand and analyze this type of phenomena. It is also intended that students acquire the necessary knowledge to further deepen knowledge in this area.

In a first phase, the essential contents of Stochastic Processes will be transmitted, in a second phase to the detailing of particular cases such as Poisson Processes and Markov Chains.

There will be an articulation between the theory of the processes and practical applications to concrete and real situations, whenever possible.

Theoretical results will be based on content exposition, demonstration of results and resolution of small examples. On the practical it will be preferred the use of computational means that allow the resolution of more complex problems. The evaluation will be done through written evaluation and practical computational work.

Evaluation method

Evaluation Rules



To obtain Frequency at UC, the student must attend at least 2/3 of the  classes.


Absences Justifications

Absences from classes must be justified until one week after the return of the student to classes.

Continuous Evaluation

Obtaining attendance is mandatory for the purposes of carrying out the assessment tests.

The assessment of knowledge of the Stochastic Processes curricular unit consists of 3 assessment elements::

  • 2 Tests, with a maximum duration of 1h30.
  • 1 Practical work, carried out in groups, during the semester.

Non-attendance in an evaluation translates into a rating of 0 values ​​in that evaluation..

Practical work may be subject to oral discussion.


Approval in Normal Season

Considering T1, T2, PW the grades obtained in Test 1, Test 2 and Practical Work, respectively, it is considered that a student passes the curricular unit if the following condition is verified:

Normal Season Grade:    NG = 0.35T1+0.35T2 + 0.3PW  >= 9.5 values.

Approval at the Time of Appeal

Any student who has attended the curricular unit may take the Appeal Exam.

The Appeal Exam Grade (AE) will integrate the student''s Resource Grade (RG) together with the Assignment (PW) grades and will be determined by: 

RG = max{0.7AE + 0.3PW , 1.0 AE}.

Grade Improvement

The student who intends to take the grade improvement exam must register, for this purpose, with the Academic Office (CLIP). Grade Improvement can be done in Resource Season. In order to improve the grade, it is required that:

  • The score obtained in this exam, IE, is not less than 9.5;
  • The Improvement Score, obtained from IS = max{0.7IE + 0.3PW , 1.0 IE} is higher than the NG score, obtained in Normal Season..

Subject matter

1. Reviews of Fundamental Concepts

2. General Notions of Stochastic Processes

3. Counting Processes

4. Discrete Time Markov Chains

5. Discrete Time Martingales


Programs where the course is taught: