# STOCHASTIC PROCESSES

7 credits during the 2nd year of 9011 Mathematics (LM-40) GENOVA

7 credits during the 1st year of 9011 Mathematics (LM-40) GENOVA

**Mathematical Statistics and Data Management 8766 (coorte 2017/2018)**- PROBABILITY 87081

**Mathematical Statistics and Data Management 8766 (coorte 2018/2019)**- PROBABILITY 87081

**Mathematical Statistics and Data Management 8766 (coorte 2016/2017)**- PROBABILITY 87081

OVERVIEW

The course presents the theory of Markov chains, both at discrete and continuous time, with particular attention to Poisson processes and queuing theory. The goal is to provide the student with the ability to model real problems of stochastic evolution in terms of Markov chains (when possible).

## AIMS AND CONTENT

LEARNING OUTCOMES

We want to introduce Markov chains and other simple stochastic processes in order to model and solve real problems of stochastic evolution.

AIMS AND LEARNING OUTCOMES

The goal is to have the student learn the language of Markov chains, so that he will be able to build an accurate model starting from real problems of stochastic evolutions taking values in a finite or countable set (the set of states).

At the end of the course the student will have to:

- know the general theory of Markov chains, both at discrete and continuous time, - classify the states and determine the invariant laws with respect to the evolutions of the system, - suitably model real situations of the queueing theory in the language of Markov chains, and be able to study the efficacy of the model.

PREREQUISITES

The basic topics on Topology, Probability.

You can find more details on Aulaweb.

Teaching methods

Teaching is done the traditional way, with lectures held at the blackboard. Expect 2 theory classes per week (4 hours) and 1 of exercises (2 hours).

At the end of the course there will be a guided full-text exercise so as to give students the opportunity to understand their degree of readiness and to clarify together possible doubts.

Attendance is not mandatory but strongly recommended.

SYLLABUS/CONTENT

*Discrete time Markov chains.* Definition. Classification of states. Transience and recurrence criteria. Probability of absorbtion in recurrence classes. Invariant laws. Limit Theorems. Convergence to equilibrium. Applications: random walks.

*Contnuous time Markov chains.* Hitting time. Chapman-Kolmogorov equations. Invariant laws. Jumps chain. Born and death chains. Poisson processes.

*Queueing theory.*

RECOMMENDED READING/BIBLIOGRAPHY

P. Baldi, Calcolo delle Probabilità e Statistica Matematica

W. Feller, An introduction to Probability Theory and its Applications

S. Karlin, H.M. Taylor, A First Course in Stochastic Processes.

S. Karlin, H.M. Taylor, A Second Course in Stochastic Processes.

S.M. Ross, Introduction to Probability Models.

G. Grimmett, D. Stirzaker, (2001). Probability and Random Processes.

J.R. Norris. Markov Chains.

P. Brémaud. Markov Chains: Gibbs Fields, Montecarlo Simulation, and Queues.

Notes

## TEACHERS AND EXAM BOARD

**Ricevimento:** Thursday: 14.00-15.30, office 836, or by arrangement made by email.

Exam Board

VERONICA UMANITA' (President)

EMANUELA SASSO

EVA RICCOMAGNO

ERNESTO DE VITO

## LESSONS

Teaching methods

Teaching is done the traditional way, with lectures held at the blackboard. Expect 2 theory classes per week (4 hours) and 1 of exercises (2 hours).

At the end of the course there will be a guided full-text exercise so as to give students the opportunity to understand their degree of readiness and to clarify together possible doubts.

Attendance is not mandatory but strongly recommended.

LESSONS START

The class will start according to the academic calendar.

## EXAMS

Exam description

Written test + oral test.

To participate in the written test you must register on the UNIGE site.

The written exam is only passed by scoring greater than or equal to 18 marks out of 30. The oral examination can be taken immediately after the written test or even in subsequent exam sessions during the academic year in progress.

The written test consists of 2 exercises, one on the discrete part and the other one on the continuous part.

The duration of the test is 3 hours and access to the course notes (including exercises done in the classroom) and handouts is allowed.

Assessment methods

The oral examination is aimed at assessing the general understanding of the course topics and it is required that the student knows how to properly expose the concepts seen in the course, to show the main results and to solve the exercises.