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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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Course ID
MAT0039
Teacher
Matteo Ruggiero
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 algorithms.

A module of the course, included in the overall courseload, will be taught by visiting professor Pierre Jacob (Harvard University, USA) on Probability couplings and Monte Carlo (cf. International visiting professors).

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## Results of learning outcomes

Knowledge and understanding: after the course, the student will have a sufficient knowledge of discrete time markov chains, continous time Markov chains, of the basic Monte Carlo sampling schemes and Markov chain Monte Carlo algorithms, and of Levy processes. The student will know the main properties of these objects and how to prove these formally.

Applying knowledge and understanding: the student will be able to manipulate the mathematical objects seen in the course in order to elicit and prove their properties, for example verifying its reversibility; finding its stationary distribution, etc. The student will also be able to construct a Markov chain Monte Carlo algorithm for a problem at hand (of comparable difficulty to those analysied in class) by choosing and appropriately adapting one of the strategies seen in class. The acquired knowledge will suffice for reading and understanding independently a research article on topics coherent with the course contents.

Making judgements: the student will be able to classify a Markov chain and a continuous time Markov chain in terms of irreducibility, periodicity, recurrence, stationarity and reversibility.

Communication skills: the student will be able to explain to a non expert the acquired concepts by using a formallly correct and rigourous exposition, and to discuss with experts about topics coherent with the course contents.

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## Program

- Introduction: stochastic processes; finite dimensional distributions; Kolmogorov's existence theorem; classes of stochastics processes based on path properties and dependence structure.

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

- Elements of Monte Carlo and Markov chain Monte Carlo methods: Monte Carlo sampling; rejection sampling; importance sampling; Markov chain Monte Carlo principle; thinning; Metropolis-Hastings algorithm; Gibbs sampler; slice sampler; mixtures and cycles of MCMC.

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

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

- Probability couplings and Monte Carlo (Pierre Jacob)

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

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## Course delivery

The course is composed of 24 lectures for a total of 48 hours. Due to Covid-19 emergency, the classes in a.y. 2019/20 will be video-recorded and uploaded on the course's Moodle page (link below).

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## Learning assessment methods

The final assessment consists in an oral examination on the material covered in class. The verification will evaluate the student's detailed knowledge and comprehension of the topics covered, together with the propriety of language, rigour and appropriate level of mathematical formality used for presenting definitions, examples, results and proofs.

During the Covid-19 emergency the final assessment will consist in a written exam with Webex video surveillance. Detailed instruction are published on the course's Moodle page (link below).

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Main references:

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

• 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

Enroll
• Open
Enrollment opening date
01/09/2019 at 00:00
Enrollment closing date
30/06/2020 at 00:00
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Last update: 14/05/2020 09:24
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