Stochastic Processes and Applications


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

Gracinda Rita Diogo Guerreiro


Weekly - 4

Total - 62

Teaching language



The student should have knowledge about Probability Theory.


Dobrow, R. (2016) Introduction to Stochastic Processes with R, John Wiley & Sons

Jones, P.; Smith, P. (2018) Stochastic Processes: An Introduction, 3rd Ed, CRC Press, Chapman & Hall

Muller, D. (2007) Processos Estocásticos e Aplicações, Almedina

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

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

Rolski, T., Schmidli, H., Schmidt, V., Teugels, J. (1999) Stochastic Processes for Insurance and Finance, Wiley

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

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

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 to the UC the student must attend 2/3 of the classes.


Absences Justifications

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


The continuous assessment will be made through two tests (T1 and T2), to be carried out face-to-face, and practical work (TP), to be carried out in groups.

Note Normal Season = 0.35 T1 + 0.35 T2 +0.30 TP, with T2> = 7 values

The student who obtains a final grade greater than or equal to 18.5 must take an oral defence of grade (on a date to be agreed). If the student does not attend the oral exam, the final grade will be 18 points.

The student obtains approval to UC if Grade Normal Season is greater than or equal to 9.5 values.



The evaluation of Appeal Season is made by Exam (E), being valid both for grade improvement and for approval to UC.

Note Appeal Season = 0.7 E + 0.3 TP, with E> = 7 values

The student obtains approval to the UC if the Grade of Appeal Season is greater than or equal to 9.5 values.

The student who obtains a final grade greater than or equal to 18.5 must take an oral defence of grade (on a date to be agreed). If the student does not attend the oral exam, the final grade will be 18 points.


Assessment Assistance Instruments

During the realization of the face-to-face assessment elements (Tests and Exams), students will be able to use a calculating machine and will be able to consult the UC Form. The Form will be provided in pdf at the beginning of classes and, on the day of the evaluations, it will be provided by the teacher on paper.



Students who intend to take the appeal exam, with a view to improving their grades, must, in advance, request this improvement from academic services.

Practical work cannot be improved.

Subject matter

0. Reviews on Fundamental Concepts

1. General notions of Stochastic Processes

1.1 Definitions

1.2 Stationary Processes and Evolutionary Processes

1.3 Processes of Independent and Stationary Increments

1.4 Random Walks

1.5 Poisson Processes

1.6 Markov Processes

2. Counting Processes

2.1 Definition

2.2 Axiomatic of the Homogeneous Poisson Process

2.3 Processes derived from the Poisson process

2.3.1 Non Homogeneous Poisson Process

2.3.2 Mixed Poisson Process

2.3.3 Composite Poisson Process

2.4 Time Between Arrivals and Waiting Times

3. Discrete Time Markov Chains

3.1 Definitions

3.2 Transition Probabilities and Chapman-Kolmogorov Equation

3.3 Classification of States and Chain Decomposition

3.4 Time of Occupation and 1st Passage

3.5 Stationary Distribution

3.6 Limit Theorems

4. Discrete Time Martingales

4.1 Definitions

4.1.1 Filtration

4.1.2 Martingala

4.2 Martingales

4.2.1 Discrete Time Martingales

4.2.2 Increments of Martingala

4.2.3 Supermartingale and Submartingale

4.3 Stopping Time

4.4 Convergence in Martingales