# 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 StatisticsPart 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)-