Stochastic Processes and Applications
Objectives
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
Code
12233
Credits
6.0
Responsible teacher
Luís Pedro Carneiro Ramos
Hours
Weekly - 4
Total - 59
Teaching language
Português
Prerequisites
The student should have knowledge about Probability Theory.
The student should have skiulls on programming (R or Python)
Bibliography
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
Available soon
Evaluation method
Evaluation Rules
Frequency
To obtain Frequency to the UC the student must aperform the Practical Assignment and the correspondent discussion.
CONTINUOUS EVALUATION / NORMAL SEASON
The continuous assessment will be made through one test (T), one mini-test (MT) to be carried out face-to-face, and a computacional assignment (TP), to be carried out in groups.
Note Normal Season = 0.6 T + 0.2 MT + 0.4 TP, with T> = 7.5 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.
The student obtains approval to UC if Grade Normal Season is greater than or equal to 9.5 values.
EVALUATION OF APPEAL SEASON
The evaluation of Appeal Season is made by Exam (E), being valid both for grade improvement and for approval to UC.
Note Appeal Season = 0.6 E + 0.4 TP, with E> = 7.5 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.
GRADE IMPROVEMENT
Students who intend to take the appeal exam, with a view to improving their grades, must, in advance, request this improvement from academic services.
The Practical Computational Assignment 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
Programs
Programs where the course is taught: