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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 2015/2016

Course ID
MAT0039
Teaching staff
Matteo Ruggiero
Andreas Kyprianou
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 Lecture
Language
English
Attendance
Optional
Type of examination
Oral
Prerequisites
PROBABILITY THEORY (MAT0034)
Propedeutic for
BAYESIAN NONPARAMETRIC STATISTICS (MAT0042)
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## Course objectives

The course introduces to the theory of Markov and Lévy processes, providing the necessary tools for modern temporal modelling in view of statistical applications.

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## Results of learning outcomes

The student will learn the structure and properties of some classes of stochastic processes widely used in applied probability and statistical inference. This includes analysing and manipulating their main features, computing the most relevant quantites of interest, and modelling stylized observed phenomena by choosing the correct type of process based on their characteristics.

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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;
• an additional topic previously agreed with the teacher (optional).

The latter can consist in the presentation of a scientific paper whose content is coherent with the course's syllabus, or the presentation of a written essay in which the student investigates in more detail a topic of interest.

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## Program

- Introduction: 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. Examples: random walks, birth and death chains, branching processes, Wright-Fisher models.

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

- Continuous time Markov chains: transition functions and Chapman-Kolmogorov equations; transition rates and infinitesimal generators; backward and forward equations; embedded chains and holding times; uniformisation; stationarity; reversibility. Examples: Poisson process, birth and death processes, Wright-Fisher models, coalescent processes.

Additionally, a 16 hours module will be taught by visiting professor Andreas Kyprianou (University of Bath, UK) on Lévy processes and Poisson random measures.

## 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.

• 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

DaysTimeClassroom
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Last update: 05/05/2017 14:41
Location: https://www.master-sds.unito.it/robots.html