Degree | Type | Year | Semester |
---|---|---|---|
2500149 Mathematics | OT | 4 | 2 |
As a general requirement, in order to be able to follow this subject, good knowledge is required at the practical level of analysis and calculus or, more specifically, integration and series. As more specific requirements, it is necessary to have previously studied a course in Probability and Stochastic Modeling.
The aim of this subject is, on the one hand, to introduce the student in the part of the theory of probability called theory of stochastic processes, whose purpose is to study the random phenomena that evolve over time or in the space We will see the basic generalities of these models and we will study some specific models.
The discrete Markov chains will be studied in general and in the particular case of the random walk. We will also study the continuous-time Markov chains, such as the Poisson process or the birth and death processes. Finally we will also introduce the Brownian motion.
1. Discrete-time Markov chains.
1.1. Motivation: the random walk.
1.2. Definitions. Basic properties. Transition matrix.
1.3. Stoping time. Strong Markov property.
1.4. Recurrence and transience.
1.5. Asymptotic behavior. Invariant distribution
1.6. Ergodic theorem
1.7. More aspects about random walk
2. Continuous-time Markov chains.
2.1. Motivation: the Poisson process.
2.2. Basic properties. Generating matrix. Differerential equations of Kolmogorov.
2.3. Invariant distribution
2.4. Ergodic theorem
2.5. More aspects of the Poisson process.
3. The Brownian motion.
This subject is semiannual and consists of two hours of theory and one hour of problems per week. There will also be three two-hour seminars.
In theory classes, the teacher plays the main role and we will work with the structure definition-theorem-proof-application.
In the problem classes students will leave on the board to solve the problems. Problem solving will be monitored at all times by the teacher.
On the other hand, in the seminar sessions, the student will work, under the tutelage of the teacher, some practical situations that are related to what has been studied in the theory classes.
These sessions will also allow, students and teachers, to be aware of the evolution in the achievement of the concepts and methods that are introduced in theory classes.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Title | Hours | ECTS | Learning Outcomes |
---|---|---|---|
Type: Directed | |||
Problem classes | 13 | 0.52 | 1, 4, 2, 7, 6, 3 |
Theory classes | 28 | 1.12 | 1, 2, 7, 3 |
Type: Supervised | |||
Seminars | 6 | 0.24 | 1, 4, 2, 7, 6 |
Type: Autonomous | |||
Prepare exams | 20 | 0.8 | 2, 7, 3 |
Study of the theory and resolution of problems | 65 | 2.6 | 1, 4, 2, 7, 6, 3 |
There will be two partial exams during the semester. The first will take place approximately in the middle of the course and the second will take place at the end of the course.
There will also be problems deliveries. The final grade will be obtained by weighting the average of the grades obtained with a weight of 45% of each part and 10% of the delivery of problems.
The minimum grade for each section, in order not to go to recovery is 3.5.
If necessary, a recovery exam will be scheduled on the day set by the coordination as the final exam.
Title | Weighting | Hours | ECTS | Learning Outcomes |
---|---|---|---|---|
First partial exam | 45% | 4 | 0.16 | 1, 2, 7, 6, 5 |
Problems and voluntary tasks | 10% | 6 | 0.24 | 1, 4, 2, 7, 6, 3 |
Recovery exam | 100% | 4 | 0.16 | 1, 4, 2, 7, 6, 3 |
Second partial exam | 45% | 4 | 0.16 | 1, 2, 7, 6 |
No specific software is needed for this course.