Teaching code

4S00254

Academic staff

Luca Di Persio, Marco Caliari

Coordinatore

Luca Di Persio

Credits

6

Scientific Disciplinary Sector (SSD)

MAT/06 - PROBABILITY AND STATISTICS

Language

Italian

Period

I sem. dal Oct 2, 2017 al Jan 31, 2018.

Lessons timetable

Moodle

Learning outcomes

Stochastic Systems [ Applied Mathematics ]
AA 2017/20178

The Stochastic Systems course aims at giving an introduction to the basic concepts underlying the rigorous mathematical description of the temporal dynamics for random quantities.

The course prerequisites are those of a standard course in Probability, for Mathematics / Physics.

It is supposed that students are familiar with the basics Probability calculus, in the Kolmogorov assiomatisation setting, in particular with respect to the concepts of density function, probability distribution, conditional probability, conditional expectation for random variables, measure theory (basic ), characteristic functions of random variables, convrgence theorems (in measure, almost everywhere, etc.), central limit theorem and its (basic) applications, etc.

The Stochastic Systems course aims, in particular, to provide the basic concepts of: Filtered probability space, martingale processes, stopping times, Doob theorems, theory of Markov chains in discrete and continuous time (classification of states, invariant and limit,measures, ergodic theorems, etc.), basics on queues theory and an introduction to Brownian motion.

A part of the course is devoted to the computer implementation of operational concepts underlying the discussion of stochastic systems of the Markov chain type, both in discrete and continuous time.

A part of the course is dedicated to the introduction and the operational study, via computer simulations, to univariate time series.

It is important to emphasize how the Stochastic Systems course is organized in such a way that students can concretely complete and further develop their own:
° capacity of analysis, synthesis and abstraction;
° specific computational and computer skills;
° ability to understand texts, even advanced, of Mathematics in general and Applied Mathematics in particular;
• ability to develop mathematical models for physical and natural sciences, while being able to analyze its limits and actual applicability, even from a computational point of view;
° skills concerning how to develop mathematical and statistical models for the economy and financial markets;
° capacity to extract qualitative information from quantitative data;
° knowledge of programming languages or specific software.

Program

Stochastic Systems [ Applied Mathematics ] AA 2017/2018 Syllabus • Conditional Expectation ( from Chap.1 of [BMP] ) • Definitions and basic properties • Conditional expectations and conditional laws • Introduction to stochastic processes ( From Chap.1 di [BMP] ) • Filtered probability space, filtrations • Adapted stochastic process (wrt a given filtration) • Martingale (first definitions and examples: Markov chains) • Kolmogorov characterization theorem • Stopping times • Martingale ( From Chap.3 of [BMP] • Definition of martingale process, resp. super, resp. lower, martingale • Fundamental properties • Stopping times for martingale processes • Convergence theorems for martingales • Markov chains (MC) ( From Chap.4 of [Beichelet] , Chap.5 of di [Baldi] ) • Transition matrix for a MC • Construction and existence for MC • Omogeneous MC (with respect to time and space) • Canonical MC • Classification of states for a given MC ( and associated classes ) • Chapman-Kolmogorov equation • Recurrent, resp. transient, states ( classification criteria ) • Irriducible and recurrent chains • Invariant (stationary) measures, ergodic measures, limit measures ( Ergodic theorem ) • Birth and death processes (discrete time) • Continuous time MC ( From Chap.5 of [Beichelt] ) • Basic definitions • Chapman-Kolmogorov equations • Absolute and stationary distributions • States classifications • Probability and rates of transition • Kolmogorov differential equations • Stationary laws • Birth and death processes ( first steps in continuous time ) • Queuing theory (first steps in continuous time) • Point, Counting and Poisson Processes ( From Chap.3 of [Beichelt] ) • Basic definitions and properties • Stochastic point processes (SPP) and Stochastic Counting Processes (SCP) • Marked SPP • Stationarity, intensity and composition for SPP and SCP • Homogeneous Poisson Processes (HPP) • Non Homogeneous Poisson Processes (nHPP) • Mixed Poisson Processes (MPP) • Birth and Death processes (B&D) ( From Chap.5 of [Beichelt] ) • Birth processes • Death processes • B&D processes ° Time-dependent state probabilities ° Stationary state probabilities ° Inhomogeneous B&D processes Bibliography [Baldi] P. Baldi, Calcolo delle Probabilità, McGraw-Hill Edizioni (Ed. 01/2007) [Beichelt] F. Beichelt, Stochastic Processes in Science, Engineering and Finance, Chapman & Hall/CRC, Taylor & Francis group, (Ed. 2006) [BPM] P. Baldi, L. Matzliak and P. Priouret, Martingales and Markov Chains – Solve Exercises and Elements of Theory, Chapman & Hall/CRC (English edition, 2002) Further interesting books are: N. Pintacuda, Catene di Markov, Edizioni ETS (ed. 2000) Brémaud, P., Markov Chains. Gibbs Fields, Monte Carlo Simulation, and Queues, Texts in Applied Mathematics, 31. Springer-Verlag, New York, 1999 Duflo, M., Random Iterative Models, Applications of Mathematics, 34. SpringerVerlag, Berlin, 1997 Durrett, R., Probability: Theory and Examples, Wadsworth and Brooks, Pacific Grove CA, 1991 Grimmett, G. R. and Stirzaker, D. R., Probability and Random Processes. Solved Problems. Second edition. The Clarendon Press, Oxford University Press, New York, 1991 Hoel, P. G., Port, S. C. and Stone, C. J., Introduction to Stochastic Processes, Houghton Mifflin, Boston, 1972

Bibliografia

Examination Methods