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Procesos Estocásticos
- Prácticas de Aula/Semina (7 Hours)
- Prácticas de Laboratorio (7 Hours)
- Tutorías Grupales (4 Hours)
- Clases Expositivas (42 Hours)
The course on Stochastic Processes is one of the optinal courses within the degree of Mathematics.
It uses the contents on Probability Theory imparted in the subject Probability and Statistics from the second year (Modulus Probability and Statistics) and in those of Measure Theory that belong to the subject Mathematical Analysis III of the third year (Modulus Advanced Mathematical Analysis).
It is related to the subjects Mathematical Analysis I and II (basic and compulsory subjects, respectively, within the Module Real and Complex Variables), Topology I (compulsory, Modulus Topology and Differential Geometry), Statistical Inference (compulsory, Modulus Probability and Statistics) and Algebra II (compulsory, Modulus Algebraic Structures).
Nowadays, Stochastic Processes constitute one of the most important tools within Probability and Statistics, and they have applications in all fields. In the last years, students from the Science Faculty of the University of Oviedo have dealt with them in their professional activities, in fields as various as Biology or Banking. Moreover, fields as apparently appart from Mathematics as Sociology or unemployment policies, and in others such as Informatics or Industrial Design their importance is growing considerably. In addition, leading studies such as Master Programs in Economics or Engineering are exploiting techniques of Stochastic Processes in matters such as market predictions or engineering reliability.
The goal of this subject is that the student becomes acquainted with the potential that stochastic processes have in modelling of phenomena, and that he becomes capacble of applying them in real life problems, establishing the basis for future studies in the most relevant areas.
This entails understanding the generic process of modelling and being capable of handling both those discrete and continuous models that are of interest in practice, including simulations, adjustment to real data, and problem resolution in Economy, Informatics, Social Sciences, etc.
Nowadays there are courses on Stochastic processes that are either a compilation of discrete processes and a basic introduction to conitnuous ones (basically Markov chains and processes), techincally rigorous courses about continuous time processes, that barely deal with the Brownian Motion, or either courses on cotninuous time processes that are less rigorous technically but that get to deal with the Brownian Motion and the Poisson Processes.
It seems thus impossible to have a course where all the models are studied, even less so with a complete theoretical study. The idea of this subject is to establish a solid, but not too complex, basis about the modelling process (transition probabilities, product spaces), as well as the techniques required to solve the problems that arise in the continuous case (separable processes), allowing the student to get the general structure that is applicable for any phenomenon (including those outside the contents of the course), and also to handle quickly the most important applied models, experimenting thus with real-life applications.
The student will become thus familiar with the most important discrete and continuous processes. In the first case, the main model are Markov Chains. In the second, it is the Brownian Motion, and more generally the Gaussian Processes.
A basic knowledge of the concepts and tools from Probability Theory and Statistics and from Measure Theory imparted in the courses Probability and Statistics from the second year and Mathematical Analysis III from the third year is assumed.
It is recommended to be proficient in the handling of probability spaces, random variables, and uni- and bivariate integration. It is also advisable to know the most important results about the asympotic behaviour of sequences of random variables and of limits of sequences of events, the extension of probability measures to greater domains and the techniques in the proofs of properties of sigma-fields from some of their subclasses.
General competences from the Degree on Mathematics:
- CG 1 (MAT). Apply knowledge acquired to his/her work in a professional manner.
- CG 2 (MAT). Elaborate and support arguments.
- CG 3 (MAT). Structure and solve problems.
- CG 4 (MAT). Collect and interpret data, information and relevant results, extract conclusions and give reasoned reports.
- CG 5 (MAT). Transmit information, ideas, problems and solutions from the mathematical field to both a specialised and a non-specialised audience.
- CG 6 (MAT). Apply the knowledge acquired and the ability of abstract analysis to the definition and structuring of problems and to the quest for solutions, both in an academic and in a professional context.
- CG 7 (MAT). Comunicate, both in a written and in a oral manner, the information, procedures, results and ideas, both to a specialised and a non-specialided audience.
- CG8 (MAT). Study and learn in an autonomous way, organizing and planning the available resources, new techniques and contents in any scientific or technological discipline.
Specific competences from the Degree on Mathematics:
- CE 1 (MAT). Understand and use the mathematical language.
- CE 2 (MAT). Acquire the ability to state propositions in different areas of Mathematics and to make proofs.
- CE 3 (MAT). Know rigorous proofs of some classic theorems in different areas of Mathematics.
- CE 4 (MAT). Assimilate the definition of a new mathematical object to other existing ones, and be able to use it in different contexts.
- CE 5 (MAT). Extract the structural properties (of mathematical objects, of the observed reality, and of other subjects), distinguishing them from other accesory properties, and prove/disprove them mathematically by means of proofs and counterexamples, being able to identify incorrect reasonings.
- CE 6 (MAT). Solve mathematical problems by means of basic calculus and other techniques.
- CE 7 (MAT). Propose, analyze, validate and interpret models for real-life scenarios, using the most appropriate mathematical techniques.
- CE 8 (MAT). Plan the solution of a problem in terms of the available tools and the existing time and resource constraints.
- CE 9 (MAT). Use specific software for statistical analysis, numerical and symbolic calculus, graph visualization, optimization, etc, as a tool for problem solving.
- CE 10 (MAT). Develop programs that solve specific mathematical problems using the most adequate computational tool.
Cross-curricular competences from the degree on Mathematics:
- CT 1 (MAT). Use bibliography and search tools that are specific to mathematical contents.
- CT 2 (MAT). Manage the timetable in an optimal way and organize the available resources, establishing priorities, alternative approaxches and identifying the logical errors in a decision process.
- CT 3 (MAT). Verify or disprove other people's arguments.
- CT 4 (MAT). Work in interdisciplinar teams, inputing order, abstraction and logical reasoning.
- CT 5 (MAT). Read scientific texts both in the mother tongue and in other relevant languages in the scientific world, particularly English.
Learning results
The learning results upon which the acquired competences are based, are the following:
- RAPE 1. Compute probabilities associated with random phenomena.
- RAPE 2. Indentify real-life scenarios where the most important probability models arise.
- RAPE 3. Handle random variables and recognize their importance as a model for real phenomena.
- RAPE 4. Know and master the notion of independence and apply, in simple cases, the central limit theorem.
- RAPE 5. Use statistical softwarw to solve real life problems.
- RAPE 6. Determine the stochastic model of a phenomenon from its transition probabilities.
- RAPE 7. Use the notion of separable process to determine properties of a process, and to verify the separability of a process.
- RAPE 8. Use the properties of the most common stochastic processes and apply them in real-life scenarios.
The course will develop the following concepts:
Product space and transtion probabilities; discrete time processes; the problem of the continuous time processes; Kolmogorov's model for continuous phenomena; separable processes; markov chains; brownian motion; gaussian processes; modelling, simulation and study of real-life scenarios; use of statistical software.
These contents are structured in the following manner:
PART 1.- Modelling
- Transition probabilities and product spaces.
- Product probabilities in 2-dimensions.
- Product probabilities in n-dimensions.
- Ionescu-Tulcea's theorem for discrete time phenomena.
- The continuous time problem.
- Kolmogorov's approach to continuous time phenomena.
- Separable processes and Doob's solution.
PART 2.- Discrete time processes
- Simple random sampling.
- Non-identically distributed sequences.
- Markov chains.
- Particular cases: birth and death processes. Queueing systems. Renewal processes.
- Asymptotic properties.
- Verification of the adjustment of a model to the data. Applications in Engineering, Economics, Sociology, etc.
PART 3.- Continuous time processes
- Brownian motion.
- Properties of the separable brownian motion.
- Visualizing the trajectory of the Brownian Motion. Simulation.
- Gaussian processes.
- Applications. Adjustment to data from financial markets. Risk evaluation, idoneity of a portfolio.
- Poisson processes.
- Simulation and application with real data. Engineering reliability. Cost evaluation and performance.
According to the EEES, the subject is structured in on-site activities and in autonomous work from the student.
On-site activities are those where the teacher is present. They can be theoretical sessions, seminars or and problem solving sessions, computer sessions, groups tutorials, and evaluation sessions.
- Theoretical sessions: In these sessions the most important theoretical aspects of the course are developed. A rigid exposition as in a textbook will be avoided, so that the results do not appear in a prely descriptive manner (definition, theorem, proof). Instead, it will be aimed that the student understands the motivation behind a definition or the underlying meaning of a theorem. We shall start by suggesting the problems to be solved and the goals to be attained, approaching the former in a gradual manner until the latter are attained. Small examples will allow to follow the different steps in the approach to a solution. Finally, the main results from the literature shall be presented.
- Seminars or problem solving sessions: The knowledge from the theoretical sessions will be consolidated by the solving of exercises of a theoretical nature. These exercises will be proposed in the theoretical classes and the student may solve them as part of his7her autonomous work. The participation of the students shall be encouraged, and in some cases the students shall solve the exercises on the blackboard, so that their solution is debated among their peers.
- Computer sessions: The most important techniques in an applied context shall be developed in the computer sessions. Real-life scenarios shall be considered and they shall be solved using the statistical software R.
- Group tutorials: In them, the student may ask questions to the other students and the teacher. It will be enoouraged for the student to evaluate the developement of the course and to identify the main difficulties, so that more time can be spent on them.
- Evaluation sessions: In them some exams where the level of knowledge acquired by the students is evaluated will take place (see next Section 7).
In order to satisfy the principles of the ECTS (R.D. 1393/2007), the student shall carry out in paraller an autonomous work, supervised by the teachers.
The distribution of the hours and of the ECTS for the on-site activities and the autonomous work for the average sudent is given in the following table:
TYPE OF ACTIVITY | Hours | ECTS | Percentage | |
On site | Theoretical classes | 40 | 1.6 | 26.7 % |
Problem solving sessions/ Seminars | 7 | 0.28 | 4.7 % | |
Computer sessions | 7 | 0.28 | 4.7 % | |
Group tutorials | 4 | 0.16 | 2.7 % | |
Evaluation sessions | 2 | 0.08 | 1.3 % | |
Total | 60 | 2.4 | 40 % | |
Autonomous | Theoretical study | 30 | 1.2 | 20 % |
Problem solving | 50 | 2 | 33.3 % | |
Computer sessions work | 10 | 0.4 | 6.6 % | |
Total | 90 | 3.6 | 60 % | |
Total | 150 | 6 | 100 % |
Working plan (orientative)
ON SITE WORK | AUTONOMOUS WORK | |||||||
Part | Theoretical class | Problem solving session/seminar | Computer session | Group tutorial | Evaluation session | On site work-total | Individual/group work | Total |
1. Modelling | 20 | 4 | 2 | 26 | 40 | 66 | ||
2. Discrete time processes | 10 | 2 | 4 | 1 | 17 | 30 | 47 | |
3. Continuous time processes | 10 | 1 | 3 | 1 | 15 | 20 | 35 | |
Evaluation | 2 | 2 | 2 | |||||
Total | 40 | 7 | 7 | 4 | 2 | 60 | 90 | 150 |
Exceptionally, if the sanitary conditions request it, other types of online teaching shall be implemented. In that case, students will be informed of the changes put in place.
As in any other subject, the student may pass the course on Stochastic Processes at the end of the semester or in one of the subsequent resists. The goal of the evaluation process is to verify if the student has acquired the expected competences and learning results.
First evaluation.-
(A) The bulk of the evaluation lies in the exercises that the student should make for the problem solving sessions. These exercises will be part of the autonomous work of the students. The amount and quality of the exercises put forward will be taken into account. The teacher will check them and may ask the student to explain the exercise during some of the problem solving sessions, so that the solution is debated among the group. The teacher may also put forward some additional exercises during the problem solving sessions. Attendance to the different sessions of the subject is necessary, and should be in any case not smaller than 90%.
Evaluation system | Competences |
Exercises of a theoretical/applied nature | CT2, CE1, CE2, CE3, CE4, CE5, CE6, CE7 |
(B) The student should also solve exercises similar to those in the computer sessions in a computer exam, using the statistical software R. This exam shall take place in the last programmed computer session.
Evaluation system | Competences |
Computer exam | CT2, CE7, CE8, CE9, CE10 |
(C) Finally, the participation of the students in the theoretical classes, problem solving sessions and group tutorials will be taken into account. The teacher will evaluate the degree of maturity in the subject that is evidenced by the comments and questions from the students.
Each part (A), (B) and (C) will be marked betwen 0 and 10. The final mark will be:
0.75 A + 0.15 B + 0.10 C
Evaluation system | Competences |
Participation in the classes | CT3, CE7 |
The weight of the different evaluation systems is given in the following table:
Evaluation system | Weight |
Theoretical/practical exams | 0% |
Exercises, projects and presentations carried out during the course | 75% |
Computer sessions | 15% |
Participation in the course activities | 10% |
Resit of July-
(D) There will be a written exam with exercises similar to those from the problem solving sessions.
Evaluation system | Competences |
Exam | CT2, CE1, CE2, CE3, CE4, CE5, CE6, CE7 |
(E) There will be a computer exam with exercises similar to those in the problem solving sessions.
Evaluation system | Competences |
Computer exam | CT2, CE7, CE8, CE9, CE10 |
The marks A and C obtained during the course are preserved (although it will be possible to repeat the work giving rise to mark A). The final mark is:
0.15 A + 0.10 C + 0.60 D + 0.15 E
The weight of the different evaluation systems is given in the following table:
Evaluation system | Weight |
Theoretical/practical exams | 60% |
Exercises, projects and presentations carried out during the course | 15% |
Computer sessions | 15% |
Participation in the course activities | 10% |
January resit.-
There will be two tests similar to those described in (D) and (E) in the July resit. In addition,
(F) The student will make a portfolio of exercises similar to those done during the problem solving sessions.
Evaluation system | Competencias |
Portfolio of exercises | CT2, CE1, CE2, CE3, CE4, CE5, CE6, CE7 |
The mark obtained in this part will be denoted by F. The mark in this resit will be given by:
0.70 D + 0.20 E+0.10 F
The weight of the different evaluation systems is:
The weight of the different evaluation systems is given in the following table:
Evaluation system | Weight |
Theoretical/practical exams | 70% |
Exercises, projects and presentations carried out during the course | 10% |
Computer sessions | 20% |
Participation in the course activities | 0% |
Students in extraordinary circumstances, differentiated evaluation, Erasmus.-
Students allowed by the Faculty to use the differentiated assessments, as well as those Erasmus students that cannot take part in the on-site activites and cannot thus be evaluated according to the parts (A), (B) and (C) by being abroad, will be evaluated by means of the two exams described in the January resist.
In the exercise sheets and scripts for the computer sessions in the virtual campus these students may find questions similar to those that will be in these exams.
The basic bibliography of this course is:
Real Analysis and Probability. Ash, R. B. Academic Press.
Probability and Measure. Billingsley, P. Wiley.
Additional resources may be found in:
Topics in Stochastics Processes. Ash, R. B., Gardner, M. F. Academic Press.
Stochastic Processes. Bass, R.F. Cambridge University Press.
Financial Calculus. Baxter, M., Rennie, A. Cambridge University Press.
Stochastic Processes with applications. Bhattacharya, R.N, Waymire, E.C. Wiley.
Probability and Random Processes. Grimmett, G.R., Stirzaker, D.R. Oxford University Press.
Bases Mathématiques du Calcul des Probabilités. Neveu, J. Masson.
A course in Statistics with R. Tattar, P., Ramaiah, S., Manjunath, B.G. Alpha Science International.