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Department of Mathematics "Giuseppe Peano"

# Laurea Magistrale (M.Sc.) in Stochastics and Data Science

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## Stochastic modelling for statistical applications

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### Academic year 2016/2017

Course ID
MAT0039
Teaching staff
Prof. Matteo Ruggiero
Prof. Dario Spano'
Year
1st year
Teaching period
Second semester
Type
D.M. 270 TAF B - Distinctive
Credits/Recognition
6
Course disciplinary sector (SSD)
MAT/06 - probabilita' e statistica matematica
Delivery
Class Lectures
Language
English
Attendance
Optional
Type of examination
Oral
Prerequisites
PROBABILITY THEORY (MAT0034)
Propedeutic for
BAYESIAN STATISTICS (MAT0070)
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## Course objectives

The course introduces to the theory of Markov chains, in discrete and continuous time, and Lévy processes. These are nowadays considered essential probabilistic instruments which should be part of a modern statistician's toolbox. As an illustrative application, some time will be devoted to introduce the basics of Markov chain Monte Carlo methods, with a few examples of the most widely used strategies.

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## Results of learning outcomes

The student will possess a quite detailed knowledge of Markov chain theory in discrete and continuous time, knowing how to formulate a model relative to the required task or application and how to analyse its properties, and will have acquired sufficient familiarity with Levy processes and Markov chain Monte Carlo methods to be able to autonomously comprehend a scientific paper on those topics.

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## Course delivery

The course is composed of 48 hours of class lectures.

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## Learning assessment methods

The final assessment consists in an oral examination on the material covered during the course.

The possibility of presenting a scientific paper whose content is coherent with the course's syllabus will be discussed at the beginning of the course.

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## Program

- Introduction: stochastic processes; finite dimensional distributions; existence theorem; classes of stochastics processes based on path properties.

- Markov chains: transition matrices, Chapman-Kolmogorov equations, strong Markov property, classification of states, invariant measures, reversibility, convergence to equilibrium.

- Elements of Markov chain Monte Carlo methods: Monte Carlo principle; Markov chain Monte Carlo principle; Metropolis-Hastings algorithm; Gibbs sampler; slice sampler.

- Continuous time Markov chains: transition functions and Chapman-Kolmogorov equations; transition rates and infinitesimal generators; backward and forward Kolmogorov equations; embedded chains and holding times; uniformisation; stationarity; reversibility; scaling limits and diffusion approximations.

- Levy processes: definition; infinite divisibility; Levy-Khintchine formula; Levy-Ito decomposition; Poisson random measures.

The material introduced will be throughly discussed and illustrated with numerous examples.

An 8 hours module, included in the courseload, will be taught by visiting professor Dario Spanò on "Introduction to stochastic modelling in Population Genetics".

## Suggested readings and bibliography

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Main references:

• NORRIS, J.R. Markov chains. Cambridge Series in Statistical and Probabilistic Mathematics.
• BREMAUD, P. Markov Chains. Springer.

Further suggested readings:

• BILLINGSLEY, P. Probability and measure. Wiley.
• GRIMMETT, G.R. and STIRZAKER, D.R. Probability and random processes. Oxford University Press.
• KARLIN and TAYLOR. A first Course in Stochastic Processes. Academic Press.
• KARLIN and TAYLOR. A second Course in Stochastic Processes. Academic Press.

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## Class schedule

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Last update: 28/06/2017 10:23
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