ECTS credits ECTS credits: 5
ECTS Hours Rules/Memories Student's work ECTS: 85 Hours of tutorials: 5 Expository Class: 20 Interactive Classroom: 15 Total: 125
Use languages Spanish, Galician
Type: Ordinary subject Master’s Degree RD 1393/2007 - 822/2021
Departments: External department linked to the degrees, Statistics, Mathematical Analysis and Optimisation
Areas: Área externa M.U en Técnicas Estatísticas (2ªed), Statistics and Operations Research
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
The objective of the course is for students to acquire a general knowledge of Stochastic Processes, through the study of typical processes, their applications in the modelling of random phenomena and as a probability tool for Statistics.
TOPIC 1. INTRODUCTION TO STOCHASTIC PROCESSES
1.1 Definition and basic concepts
1.2 Basic types of processes
1.3 Two important examples: the Poisson process and Brownian motion
TOPIC 2. DISCRETE-TIME MARKOV CHAINS
2.1 Basic Definitions and Properties
2.2 Transition Probabilities. Chapman–Kolmogorov equations
2.3 Classification of States
2.4 Existence of the stationary distribution and convergence theorems
2.5 Detailed Equilibrium Condition
TOPIC 3. CONTINUOUS-TIME MARKOV CHAINS
3.1 Definition and basic properties. Examples: Poisson processes, birth and death processes, multi-state models
3.2 Instantaneous jump rates and Kolmogorov equations
3.3 Asymptotic behavior. Detailed equilibrium condition
TOPIC 4. MARTINGALES
4.1 Elements of Probability and Conditional Expectation
4.2 Definition of martingale
4.3 Basic Properties
4.4 Optional Stopping Time Theorem
4.5 Convergence of martingales
4.6 Martingales in continuous time
TOPIC 5. BROWNIAN MOTION
5.1 Brownian motion: motivation and definition
5.2 Basic Properties
5.3 Simulation of Brownian motion
5.4 Properties of Brownian motion as a martingale
5.5 Markovian properties of Brownian motion. The principle of reflection
TOPIC 6. CONVERGENCE OF STOCHASTIC PROCESSES
6.1 Reminder of convergence in the distribution of random variables
6.2 Convergence in distribution in metric spaces
6.3 Notable examples: Euclidean space and C-space in [0,1]
6.4 Relative compactness and tightness. Prohorov's Theorem
6.5 Skorohod space, D[0,1]
6.6 Donsker's theorem
6.7 Convergence of empirical processes
TOPIC 7. STOCHASTIC INTEGRATION
7.1 Definition of the Itô integral
7.2 Basic Properties
7.3 Itô Formula and Applications
TOPIC 8. STOCHASTIC DIFFERENTIAL EQUATIONS
8.1 General Model and Notable Examples of Stochastic Differential Equations
8.2 Simulation of stochastic differential equations
8.2 Estimation of stochastic differential equations
Basic bibliography
BILLINGSLEY, P. (1999). Convergence of Probability Measures (second edition). Wiley.
DURRETT, R. (2012). Essentials of Stochastic Processes (second edition). Springer.
IACUS, S.M. (2008). Simulation and inference for stochastic differential equations. Springer.
Complementary bibliography
BASS, R.F. (2011). Stochastic Processes. Cambridge University Press.
BATH, U. N. (2002). Elements of Applied Stochastic Processes (third edition). John Wiley & Sons.
BATTACHARYA, R.N. y WAYMIRE, E.C. (2009). Stochastic Processes with Applications (revised edition). Siam.
GRINSTEAD, C.M. y SNELL, J.L. (1997). Introduction to Probability. American Mathematical Society.
KARLIN, S. y TAYLOR, H.M. (1975). A First Course in Stochastic Processes. Academic Press.
KARLIN, S. y TAYLOR, H.M. (1981). A Second Course in Stochastic Processes. Academic Press.
KULKARNI, V.G. (2010). Modelling and Analysis of Stochastic Systems (second edition). Chapman & Hall.
MIKOSCH, T. (1998). Elementary Stochastic Calculus, with Finance in View. World Scientific Publishing.
MÖRTERS, P., & PERES, Y. (2010). Brownian Motion. Wiley.
ROSS, S.M. (1996). Stochastic Processes (2nd Edition). John Wiley & Sons.
STEELE, J.M. (2001). Stochastic Calculus and Financial Applications. Springer.
WILLIAMS, D. (1991). Probability with Martingales. Cambridge University Press.
In this subject, the basic, general and transversal competencies included in the degree report will be worked on. The specific competencies that will be enhanced in this area are indicated below:
[SC1] Know, identify, model, study and solve complex problems of statistics and operations research, in a scientific, technological or professional context, arising in real applications.
[CE3] Acquire advanced knowledge of the theoretical foundations underlying the different methodologies of statistics and operations research, which will allow their specialized professional development.
[CE4] Acquire the necessary skills in the theoretical-practical management of probability theory and random variables that allow their professional development in the scientific/academic, technological or specialized and multidisciplinary professional field.
[CE5] Deepen knowledge of the specialized theoretical-practical foundations of modeling and study of different types of dependency relationships between statistical variables.
[CE6] Acquire advanced theoretical and practical knowledge of different mathematical techniques, specifically aimed at assisting in decision-making, and develop the capacity for reflection to evaluate and decide between different perspectives in complex contexts.
[CE8] Acquire advanced theoretical and practical knowledge of the techniques used to make inferences and contrasts related to variables and parameters of a statistical model, and know how to apply them with sufficient autonomy in a scientific, technological or professional context.
[CE10] Acquire advanced knowledge about methodologies for obtaining and processing data from different sources, such as surveys, the internet, or "cloud" environments.
The face-to-face activity of the students will be 35 hours between expository and interactive teaching. In the expository part, the teaching staff will make use of multimedia presentations, while in the interactive part the students will solve different questions raised about the contents of the subject. Some standard problems will also be solved, so that students can work on the exercise reports that will be provided. In class, some examples of simulation will be developed using the R package.
Regarding the material for monitoring the subject, in addition to the recommended bibliography, students will have teaching material prepared for the subject available through the master's degree web platform.
In accordance with the organization of the expository and interactive sessions according to the topics (see section on teaching methodology), the assessment of learning will be carried out as follows:
- Continuous evaluation (exercises, questions, small projects): 40%
- Written exam : 60%
In the second opportunity for evaluation (retake), an exam will be carried out and the final grade will be the maximum of three amounts: the mark of the ordinary evaluation, the grade of the new exam, and the weighted average of the new exam and the continuous evaluation.
Presentation to the evaluation: the student is considered to attend a call when he or she participates in activities that allow him or her to obtain at least 50% of the final evaluation.
Core and transversal competencies are assessed both in the continuous assessment processes and in the exam. The general competencies CG1, CG2, CG4 and CG5, the basic competencies CB6, CB7 and CB9 and the transversal competencies CT1 and CT3 are assessed in the exam and in the continuous assessment, while the general competence CG3, the basic competencies CB8 and CB10 and the transversal competencies CT4 and CT5 are evaluated in the continuous assessment. Of the specific competencies, both the continuous assessment and the examination address the CE1, CE3, CE4, CE5, CE6, CE8 competencies, while the continuous assessment addresses the CE10 competence.
The amount of work required to pass this subject obviously depends on the student's skills and abilities, as well as their knowledge of probability. In general, about 1.5 hours for each expository session (for concept review and bibliographic consultation) and one hour for each hour of interactive teaching should be sufficient. For the exercises that will be proposed, 10 hours of personal work are considered. For the final exam (completing the assessment), 3 hours are counted.
Attendance at the expository and interactive sessions is essential for the follow-up and improvement of the subject. Students must complete all the activities recommended by the teaching staff (problem solving, literature review and practical exercises) in order to successfully complete the subject.
It is reported that the contents of this subject include probability proofs with high mathematical content. It is therefore recommended to attend the subject with a high level of skill and interest in mathematical results related to Probability.
The development of the contents of the subject will be carried out taking into account that the competencies to be acquired by the students must comply with the MECES3 level. The contents included in this subject are technically advanced and will be analysed with an eminently theoretical approach, although some practical applications will be presented.
In cases of fraudulent completion of exercises or tests, the provisions of the respective regulations of the universities participating in the Master Degree on Statistical Techniques will apply.
This guide and the criteria and methodologies described in it are subject to modifications derived from the regulations and guidelines of the universities participating in the Master Degree on Statistical Techniques.
Cesar Andres Sanchez Sellero
Coordinador/a- Department
- Statistics, Mathematical Analysis and Optimisation
- Area
- Statistics and Operations Research
- Phone
- 881813208
- cesar.sanchez [at] usc.es
- Category
- Professor: University Lecturer
| 05.20.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 04 |
| 07.04.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 04 |