• Programmi dei corsi

    Lo scopo dei corsi del primo anno è fornire una base comune agli studenti provenienti da diverse lauree magistrali, tipicamente in Economia, Statistica, Matematica, Fisica o Ingegneria. I corsi comuni a entrambi i curricula sono i seguenti:

    ANALYSIS

    • Sequences and series of functions.
    • Normed, metric and topological spaces.
    • Measure theory, Lebesgue integral, Lebesgue-Stiltjes integral.
    • Beppo Levi’s Theorem, Dominated Convergence, Fatou Lemma.
    • Differentiation and integration.
    • L^p spaces and basic notions on Hilbert spaces.

    Rudin, W. Principles of Mathematical Analysis, McGraw-Hill International 1976
    Rudin, W. Real and Complex Analysis, McGraw-Hill International 1986


    PROBABILITY 

    • Probability spaces, and sigma-fields. Independent events and conditional probability.
    • Random variables, expected values and main inequalities.
    • Types of convergence of random variables. Law of large numbers and central limiti theorem.
    • Introduction to stochastic processes: Markov chains, Martingales, Poisson process and Brownian motion.

    Billingsley. Probability and Measure, Wiley.
    Williams. Probability with Martingales, Cambridge University Press.


    OPTIMIZATION

    Part I: Linear Algebra

    • Eigenvalues and Eigenvectors.
    • Matrix Factorizations (Diagonalization, Spectral Decomposition, Jordan Decoposition, Schur Decomposition).
    • Generalized Inverses.

    Schott J.R., Matrix Analysis for Statistics, John Wiley & Sons, Second edition, 2005.

    Part II: Finite dimensional optimization

    • Static optimization in open and closed sets.
    • Equality and inequality constraints. Lagrange multipliers.
    • The role of convexity and its generalization in optimization.
    • Lagrangian duality.

    Bazaraa, Sheraly, Shetty, Nonlinear programming, 2008

    Part III: Numerical methods for linear algebra and optimization


    STATISTICAL INFERENCE

    Part I: Theory and Methods (36 hours)

    • Statistical models and inference. Exponential (dispersion) families. Sufficient and minimal sufficient statistics. Completeness.
    • Likelihood principle and likelihood quantities. Maximum likelihood estimators, likelihood based tests and confidence regions. Numerical aspects of likelihood procedures.
    • Asymptotic theory: rate of stochastic convergence, delta method, asymptotic approximations of likelihood procedures, Edgeworth and Cornish-Fisher expansions.
    • Pseudo-likelihoods. Estimating equations. Likelihood inference under model misspecification.
    • Penalized likelihood and model selection.
    • Elements of decision theory: optimal procedures for estimation, testing and confidence intervals.
    • Nonparametric curve estimation.

    Part II: Lab sessions with R
    Part III: Models

    • Mixed and Random Linear models. Multilevel Model.
    • Generalized Linear Models. Probit and Logit Models. Log-linear Model. Models for Ordinal Response Data.
    • Generalized Estimating Equations for Correlated Data.
    • Limited Dependent Variables Models. Truncated Variables Models.
    • Sample Selection Models.
    • Introduction to Causal Inference.
    • From Linear to Structural Models. Path Analysis.
    • Latent variables. Factor analysis. Latent Structural Models. Lisrel Model.
    • Uniqueness of solutions. Identification and Indeterminacy problems.
    • Structural Models with proxies of latent variables. Component Analysis. Partial Least Squares.
    • Latent Markov Models.

    Part IV: Lab sessions with R-SAS

    I corsi obbligatori per il Curriculum di Statistica (ma potenzialmente molto utili anche per il curriculum di Finanza Matematica) sono i seguenti:

    BAYESIAN STATISTICS

    Part I: Bayesian Statistics

    • Prior, posterior and predictive distributions.
    • One-parameter models. Normal model.
    • Gibbs sampling and MCMC methods.
    • Multivariate normal model.
    • Hierarchical models.
    • Linear model. Generalized linear models.
    • Metropolis-Hastings sampling.
    • Latent variable models.

    Part II: Lab sessions with R and specific software (WinBugs, JAGS, STAN)


    COMPUTATIONAL STATISTICS

    • Stochastic Simulations: R: fundamental and programming. Pseudo-random number, Generation from parametric Statistical Distributions, Introduction to Monte Carlo for numerical integration
    • Computer-intensive inference: Mathematica:fundamental & programming, Jacknife, Bootstrap, Bootstrap Confidence Interval, EM Algorithm
    • Introduction to Statistical Learning: Supervised and non-supervised selected techniques (with dedicated software)
    • Advanced in MCMC (with dedicated software)

    I corsi obbligatori per il curriculum di Finanza Matematica (ma potenzialmente utili anche per gli studenti del curriculum di Statistica interessati alla Finanza) sono i seguenti:

    MATHEMATICAL FINANCE

    Part I: Arbitrage and Superhedging

    Part II: Martingales, stochastic integrals and and introduction to stochastic differential equations

    Part III: Advanced topics in Option Pricing

    OPTION PRICING: FROM MONTE CARLO TO QUANTIZATION

    (corso in convenzione con il Politecnico)-