Unit  Credits  Academic sector  Period  Academic staff 

Lezione  8  MAT/05MATHEMATICAL ANALYSIS  1° sem lez 
Marco Minozzo

Esercitazione  2  MAT/05MATHEMATICAL ANALYSIS  1° sem lez 
Marco Minozzo

The course provides to students in economics and finance an overview of the theory of probability at an intermediate level.
Prerequisite to the course is an elementary knowledge of probability at the level of an undergraduate first or second year introductory course in probability and statistics.
In particular, a basic knowledge of the following topics is recommended: most common univariate discrete and continuous distributions; weak law of large numbers; central limit theorem.
The final objective of the course is to give an introduction to the advanced theory of conditional expectation, of stochastic processes in the discrete and continuous time domains and to stochastic integration.
Finite probability spaces: sample spaces; events; algebras and probability spaces; event trees; elementary conditional probability; independence; random variables; conditional probability and conditional expectation with respect to a finite partition.
Discrete time martingales on finite probability spaces: martingales and stopping times; betting strategies and impossibility of a winning betting strategy.
Infinite probability spaces and Kolmogorov’s axioms: infinite probability spaces; sigmaalgebras; random variables: discrete, absolutely continuous and singular; conditional expectation with respect to a sigmaalgebra.
Multidimensional random variables: multidimensional discrete and continuous random variables; joint distribution function; joint density function; marginal and conditional distributions; marginal and conditional densities; independence; covariance; coefficient of correlation of Bravais; CauchySchwarz inequality; joint moment generating function.
Distributions of functions of random variables: transformations of random variables; method of the distribution function; distribution of the minimum and the maximum; method of the moment generating function; transform ; probability integral transform; transformations of vectors of random variables.
Limits of random variables: infinite series of random variables; convergence in probability, in distribution, with probability one (almost surely) and in mean; weak law of large numbers and law of large numbers of Bernoulli for relative frequencies; central limit theorem; Borel’s lemma and Borel’s strong law of large numbers; order statistics; empirical distribution function.
Discrete time Markov chains: Markov processes and Markov chains; homogeneous Markov chains; transition probabilities and marginal probabilities; ChapmanKolmogorov equations; equivalence classes and irreducible chains; classification of states: transient, persistent, periodic and ergodic states; stationary distribution and ergodic theorem; Markov chain Monte Carlo and the Metropolis algorithm.
Continuous time stochastic processes: definitions and finitedimensional distributions; filtrations; adapted processes; filtrations generated by a stochastic process; stationary processes; processes with stationary increments and with independent increments; counting processes and Poisson processes; Gaussian processes and Wiener processes (Brownian motions); Wiener process as a limit of a random walk; properties and irregularities of the sample trajectories (non derivability and infinite variation); Markov processes, transition probabilities and ChapmanKolmogorov equations; continuous time martingales.
Stochastic integrals: overview of RiemannStiltjes integral; definition and properties of Itô’s integral; Itô’s formula, properties and applications; martingales associated to a Wiener process; diffusions; geometric Brownian motion and OrnsteinUhlenbeck processes.
The course consists of a series of lectures (48 hours) and exercise classes (20 hours).
All classes are essential to a proper understanding of the topics of the course.
The working language is Italian.
For the official examination both written and oral sessions are mandatory.
The course is considered completed if the candidate has done the written test and passed the oral exam.
Students that has received at least 15 out of 30 in the written exam are allowed to attend the oral exam.