Module page
Processi Stocastici
| academic year: | 2013/2014 |
| instructor: | Mauro Piccioni |
| degree courses: | Mathematics (magistrale) Mathematics for applications (magistrale) |
| type of training activity: | caratterizzante |
| credits: | 6 (48 class hours) |
| scientific sector: | MAT/06 Probabilità |
| teaching language: | italiano |
| period: | I sem (30/09/2013 - 17/01/2014) |
Lecture meeting time and location
Presence: highly recommended
Module subject: Discrete time Markov chains: definition, communicating class, absorbing state, irreducibility, adsorbing probability. Stopping times, strong Markov property, transience, recurrence. Stationarity, reversibility, periodicity, convergence to equilibrium, ergodic theorem, examples of applications. Continuous time Markov chains: definition, infinitesimal generator, backward/forward Kolmogorov equations, explosion. Poisson process, birth and death processes. Brownian motion and Gaussian processes.
Detailed module subject: Detailed course program
Suggested reading:
J.R. Norris. Markov chains. Cambridge Series in Statistical and
Probabilistic Mathematics, Cambridge, 1997.
P. Bremaud, Markov chains, Gibbs fields, Monte
Carlo simulation, and queues. Springer-Verlag, New York, 1999.
G. R. Grimmett e D. R. Stirzaker, Probability and random processes. Third edition, Oxford Univesity Press, New YOrk 2001.
Issue: Updated version of the solutions to exercises in Norris, Markov Chains
Type of course: standard
Exercise: Suggested exercises from Chapter 6, Grimmett and Stirzaker:6.1 - n. 2,3,8,9,10,11 e 12, 6.2 - 2,4, 6.3 - 1 e 3 (without mean recurrence time),2,4 (simplify the chain),5,8 e 10 (first part).
Examination tests:
- Written exam, 14/11
- second written test, 16/1/2014
- Written test of 1/23, with solutions
- Written test, 13/2/2014, with solutions
Knowledge and understanding:
Students have to acquired familiarity with Markov stochastic
processes, thought of as random dynamical systems with loss of
memory. At the end of the course, students will be able to deal
with concepts as recurrence, transience, reversibility,
stationarity, convergence to equilibrium, stopping time, Markov
property.
Skills and attributes:
Successful students will be able to model
concrete phenomena by means of Markov stochastic process, as in queuing and biological applications, to
determine the asymptotic behavior and dynamical features as
transience and recurrence, to compute possible invariant and
reversible distributions.
Personal study: the percentage of personal study required by this course is the 65% of the total.