Stochastic Processes and Modeling


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





Responsible teacher

Susana Maria Marques Henriques Botelho Baptista


Weekly - 3

Total - 48

Teaching language



Available soon


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



 Continuous assessment consists of conducting, during the semester, 1 project and 2 tests, each of which is rated from 0 to 20 points.

Let P, T 1 and T 2 be the classifications obtained in the project, 1st and 2nd tests, respectively. The student will be approved if

 0,25 × P  + 0,35×T 1  + 0,4×T 2  ≥ 9,5 .

 In this case the final classification will be given by this average rounded to the units.



 All students enrolled in the Course 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. 



 Any student wishing to perform a grade improvement must register for this purpose at CLIP (information at the Academic Office). The improvement may regard any module: 1 - Time Series, 2- Markov Chains 3 - Diffusion Processes or any combination of modules, corresponding in that case to the realization of  a project (Module 1) or an exam (Modules 2 and 3). 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 where the course is taught: