Stochastic Processes and Modeling
Objectives
Apply the concepts and properties of: stationary, integrated, univariate time series, stationary random series, filter applied to a stationary random series, backwards shift operator, backwards difference operator, roots of the characteristic equation of time series, multivariate autoregressive model.
Outline the processes of identification, estimation and diagnosis of a time series, the criteria for choosing between models and the diagnostic tests that might be applied to the residuals Develop deterministic forecasts from time series data.
Formulate the Chapman-Kolmogorov equations, calculate the stationary distribution and apply Markov chains as a tool for modeling and in simulation.
Apply the main concepts of Wiener Process. Show working understanding of stochastic differential equations, Ito integral, diffusions and mean reverting processes. Ito’s Lemma and proof, apply it to write down the stochastic differential equations for important processes and solve it.
General characterization
Code
12456
Credits
6.0
Responsible teacher
Pedro José dos Santos Palhinhas Mota
Hours
Weekly - 3
Total - 48
Teaching language
Português
Prerequisites
Available soon
Bibliography
Time Series Analysis and its applications- with R examples
Robert H Shumway and David S Stoffer
Springer Fourth Edition
Stochastic Processes
Parzen, E.
Holden Day, 1965
Stochastic Processes, 2nd Ed.
Ross, S. M.
Wiley & Sons, 1996
Stochastic Differential Equations
Oksendal, B.
Springer Sixth Edition
Statistical Inference for Diffusion Type Processes
Prakasa Rao, B.
Oxford University Press
Teaching method
The classical methodology used in Mathematics at the university level. The contents are presented and discussed trying to stress the most important ideas and practical procedures. There are study materials: text book, classroom notes with problems, some with solutions and, a list of questions indicating exactly what the student has to know and master.
Evaluation method
1. FREQUENCY
Obtaining frequency depends on the completion of 1 project.
2. CONTINUOUS EVALUATION
Continuous assessment consists of conducting, during the semester, 1 project (mandatory to obtain frequency) and 2 tests, each of which is rated from 0 to 20 points.
Let P, T1 and T2 be the classifications obtained in the project, 1st and 2nd tests, respectively. The student will be approved if
0,25 × P + 0,35×T1 + 0,4×T2 ≥ 9,5 .
In this case the final classification will be given by this average rounded to the units.
3. EXAM
All students enrolled in the Course (with frequency) can apply to the exam.
The final grade is computed according to the given formula:
FG= 0,25 × P + 0,75 × E , where P is the project classification and E the exam classification (0 to 20 points scale).
If the classification FG is higher or equal to 9.5 the student is approved with this classification, rounded to the nearest integer.
4. GRADE IMPROVEMENT
Any student wishing to perform a grade improvement must register for this purpose at CLIP (information at the Academic Office).
The improvement may regard one or more of the evaluated modules.
The improvement final classification is obtained according to 2. If this result is higher than the one already obtained, it will be taken as a final grade. Otherwise, there is no grade improvement.
Subject matter
1.Time Series
1.1. Measures; Stationary Time Series; Estimation of correlation
1.2. AR models; Ma models; ARMA models
1.3. Autocorrelation and Partial autocorrelation
1.4. ARIMA models; Building ARIMA
1.5. Predicition
1.6 Spectral representation; Periodogram
1.7 VAR
2. Markov chains in continuous time:
2.1. Homogeneous Markov processes, Kolmogorov equations
2.2. Transition Probabilities and Chapman-Kolmogorov Equation
2.3. Stationary Distribution
2.4. Non-Homogeneous Markov Processes, Matrix of Intensities, Kolmogorov Equations
2.5. Limit Theorems
2.6. Estimation
3. Diffusion Processes:
3.1. Brownian or Wiener process: construction and properties
3.2. Itô''''''''s Stochastic Integral: construction and properties; Itô''''''''s formula and applications
3.3. Stochastic Differential Equations: existence and uniqueness of strong solutions
3.4. Geometric Brownian Processes, Vasicek, Ornstein-Uhlenbeck, Cox-Ingersoll-Ross
3.5. Diffusions; essential properties in dimension one.
3.6. Estimation
Programs
Programs where the course is taught: