30216 - Probability Models

Learning outcomes

Upon completion of the course, the student will have acquired a solid understanding of the fundamental principles of discrete-time stochastic processes, with a particular focus on martingale theory, Markov chains, and stochastic control. Additionally, they will be able to analyze application-related stochastic systems using computational implementation tools

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

• Dispense del docente (Virtuale)
• Pierre Brémaud, Markov Chains (Second edition), Springer (2020).
  
• Chung and AitShalia, Elementary Probability Theory. Springer (2003)
  
• 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

Frontal lectures

Assessment methods

Oral exam

Teaching tools

Virtuale

Office hours

See the website of Salvatore Federico