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Academic Year 2023/2024 - Teacher: Vittorio ROMANO

Expected Learning Outcomes

The aim of the course is to give the main tools for statistical investigations along with the study of advanced subjects for tackling problems of interest  in mathematical physics, economics, industry and in general for applications arising from applied sciences. In particular, the course  furnishes the background for  mathematical analysis in economics and therefore suitable for student in the program of finance.  

Some aspects treated in the course are in any case relevant for those who want to teach Mathematics at the high schools.  

In particular, the course aims to allow the student to acquire the following skills:

knowledge and understanding:  knowledge of results and fundamental methods  in advanced statistics, stochastic processes and Monte Carlo method. Skill of understanding problems and to extract the major features. Skill of reading, understanding and analyzing a subject in the related literature and present it in a clear and accurate way. 

applying knowledge and understanding: skill of elaborating new example or solving novel theoretical exsercise,  looking for the most appopriate methods and applying them in an appropriate way. 

making judgements: To be able of devise proposals suited to correctly interprete complex problems in advanced statistics, stochastic processes and Monte Carlo simulation.  To be able to formulate autonomously  adequate judgements on the applicablity of simulation methods or statistical or stochastic models to theoretical or real situations. 

communication skills: skills of presenting arguments, problems, ideas and solutions in mathematical terms with clarity and accuracy and with procedures  suited for the audience, both in an oral and a written form. Skill of clearly motivating the choice of the strategy, method and contents, along with the employed computational tools.

learning skills: reading and analyzing a subject in the literature involving applied mathematics. To tackle in an autonomuous way the systematic study of arguments not previously treated.  To acquire a degree of autonomy such that the student can be able to start with an autonomuos reserach activity.

Course Structure

Mainly frontal lectures. Moreover, the theoretical acquired competencies will be applied in a laboratory where study cases will be tackled in a MATLAB environment. 

If restrictions will be introduced because the COVID pandemic, le lectures will be given in a mixed way or only online and some changes could be introduced to assure the accomplishiments foreseen for the course. 

Learning assessment may also be carried out on line, should the conditions require it.

Required Prerequisites

Calculus of probability. The attendance to the course Mathematical and statistical Methods 1 is strongly suggested. 

Attendance of Lessons

Strongly suggested. 

Detailed Course Content

  1.  Maximum likelihood method. Normal correlation. Bayesian statistical inference. Maximum entropy method: information theory of Shannon, statistical inference of distributions, application to the classical and quantum statical mechanics (deduction of the distributions of Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein). Stochastic processes: general features, processes of Poisson and counting, symmetric and non-symmetric random walk and continuous limit, Brownian motion, applications in finance, in bio-mathematics and physics. .  Stochastic differential equations and Ito's formula. Markov chains: classification of the states, absorption probability into a closed class, invariant probability. Theorem of Metropolis and simulated annealing for stochastic optimization.  Monte Carlo method: theoretical foundations, generation of random numbers, examples of applications (Bernoulli-Laplace diffusion, urn of Ehrenfest, evaluation of options in finance, Ising model). 

Textbook Information

Notes of the lecturer

V.  Romano, Metodi Matematici per i Corsi di Ingegneria, CittàStudi

P. Baldi Calcolo delle probabilità e statistica, McGraw-Hill

R. Scozzafava Incertezza e probabilità, Zanichelli

A. Rotondi, P. Pedroni, A. Pievatolo Probabilità Statistica e Simulazione, Springer

L. C. Evans, An introduction to stochastic differential equations, AMS

D. C. Montgomery, G. C. Runger Applied statistics and probability for engineers, J. Wiley 

Course Planning

 SubjectsText References
1 Metodo della massima verosimiglianza. Correlazione normale. Inferenza statistica Bayesiana. Metodo della massima entropia. Processi stocastici. Equazioni differenziali stocastiche. Metodi Monte Carlo. notes of the lecturer

Learning Assessment

Learning Assessment Procedures

Written report with applications in Matlab and oral exam.

If necessary the exam will be  online.

Examples of frequently asked questions and / or exercises

 Bayesian statistical inference. Maximum likelihood method. Method of maximum entropy. Homogeneous Markov processes. Markov chains. Absorption problems in a Markov chain.   Both and death chains. Wiener-Levy processes. Random walk. Ito formula. Generation of random numbers. More Carlo method. Metropolis algorithm. Simulated annealing. Ising model.