30216 - Probability Models

Academic Year 2023/2024

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Computer Science (cod. 5898)

Learning outcomes

At the end of the course the student knows elements of some advanced probability theories with applications to computer science, such as Markov chains with discrete and continuous time. She / he is able to analyze some simple stochastic systems related to applications.

Course contents

  • Recaps of basic probability topics in discrete spaces.

         Probability spaces, random variables, independence, measurability, expected value and conditional expected value. Convergence and law of large numbers.

  • Stochastic processes in discrete time and discrete spaces.

    General notions. Filtrations. Martingales and random walks.
  • Math finance in discrete time.

         One-period market models. Valuation and hedging of derivatives. Fundamental theorems of asset pricing. Multi-periodal models. Binomial model and extension. 

  • Markov chains.

          Introduction to Markov chains. Construction off Markov chains. State classifications. Stationary distributions and convergence.

  • Stochastic control.

        Formulation of stochastic control problems in discrete time. Dynamic programming: Bellman's equation and verification theorems. Applications.

    Readings/Bibliography

    • Lecture notes on Virtuale
    • Pierre Brémaud, Markov Chains (Second edition), Springer (2020).
    • Catene di Markov e Applicazioni Algoritmiche, Dispense del corso "Metodi probabilistici per l'informatica" del Prof. Goldwurm (UNIMI).
    • W. Woess. Catene di Markov e teoria del potenziale nel discreto. Quaderni UMI (1996).
    • A. Pascucci e W. Runggaldier, Finanza matematica. Teoria e problemi per modelli multiperiodali. Springer Unitext (2009).

    Teaching methods

    • Pierre Brémaud, Markov Chains (Second edition), Springer (2020).
    • Catene di Markov e Applicazioni Algoritmiche, Dispense del corso "Metodi probabilistici per l'informatica" del Prof. Goldwurm (UNIMI).
    • W. Woess. Catene di Markov e teoria del potenziale nel discreto. Quaderni UMI (1996).
    • A. Pascucci e W. Runggaldier, Finanza matematica. Teoria e problemi per modelli multiperiodali. Springer Unitext (2009).

    Assessment methods

    Oral exam

    Teaching tools

    Virtuale

    Office hours

    See the website of Salvatore Federico