Faculdade de Ciências e Tecnologia

Stochastic Processes

Code

3120

Academic unit

Faculdade de Ciências e Tecnologia

Department

Departamento de Matemática

Credits

6.0

Teacher in charge

Maria Fernanda de Almeida Cipriano Salvador Marques

Weekly hours

4

Teaching language

Português

Objectives

-Main objectives:
Being a mandatory course for the third year of the degree in Mathematics of FCT/UNL, in the branch Applied Mathematics, this course intends to give appropriate knowledge foundations for the study of the evolution of random phenomena.
-Main objectives related to knowledge:
Knowledge of some essential notions for the understanding of stochastic processes such as Kolmogorov''''''''''''''''s conditional expectation. Knowledge of some of the fundamental examples of discrete and continuous time stochastic processes, their main properties and examples of relevant applications.
Main objectives related to know-how:
-generic:
To acknowledge and make use of the main properties of chosen examples of stochastic processes in discrete and continuous time pertinent for applications. To be able to decide which is the more appropriate model of a stochastic process to use when faced with a realistic situation.
-specific:
To master conditional expectation properties and methods of calculation. To identify a martingale and utilize the characteristic properties of this type of processes in the study of its behaviour, in particular, in the determination of a possible asymptotic behaviour. To identify a Markov chain and utilize the characteristic properties of this type of processes for the analysis of a concrete model. Identical set of competencies for Poisson and Wiener processes.

Prerequisites

The student is strongly advised to have previously obtained good pass marks on courses covering the standard topics of measure theory and integration and probability and statistics such as the graduate courses presently offered by the department graduate program in Mathematics: Probability and Statistics I, II, Measure Integration and Probability.

Subject matter

1. Conditional expectation

2.  Kolmogorov construction of a stochastic process.

3. Discrete time martingales.

4. Markov chains.

5. Continuous  time martingales

6. Markov process.

7. Poisson process.

8. Wiener process.

9. Information Theory.

Bibliography

 

Measure theory

[1]  - Marek Capinski and Ekkehad Kopp: Measure, Integral and Probability.

Springer-Verlag

[2] - Paul Malliavin: Integration and Probability. Springer-Verlag.

Stochastic Processes

[3] – J. L. Doob:  Sochastic Processes. John Wiley and Sons.

[4] - Zdzistaw Brzeiniak and Tomasz Zastawniak: Basic Stochastic Processes. Springer

Evaluation method



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