• 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 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.

• 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

• 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)

• 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

• Opition pricing

(corso in convenzione con il Politecnico)-