81624 - PROBABILITY

Conoscenze e abilità da conseguire

As for probability, at the end of the course the student has good knowledge of probability theory of discrete and continuous random variables. Particular attention is paid to the theory of stochastic processes, both diffusive and with jumps. The student masters the main techniques of stochastic processes applied to finance, such as Ito's lemma, Girsanov theorem and change of measure methods for Lévy processes.

Contenuti

Notation and basic set theory
Sets and functions
Outer measure
Lebesgue-measurable sets and Lebesgue measure
Basic properties of Lebesgue measure
Borel sets
Lebesgue-measurable functions
Random variables
Fields generated by random variables
Probability distributions
Independence of random variables
Integral
Definition of the integral
Monotone convergence theorems
Integrable functions
The dominated convergence theorem
Relation to the Riemann integral
Approximation of measurable functions
Integration with respect to probability distributions
Absolutely continuous measures:
examples of densities
Expectation of a random variable
Characteristic function
Spaces of integrable functions
The space L
The Hilbert space L
Properties of the L -norm
Inner product spaces
Orthogonality and projections
The LP spaces: completeness
Moments
Independence
Conditional expectation (first construction)
Product measures
Independence again
Conditional probability
Strong law of large numbers
Weak convergence
Central limit theorem

Brownian Motion

Elementary Stochastic Processes

Testi/Bibliografia

Recommended texts:

[1] A Probability Path

Sidney Resnick

https://www.amazon.it/Probability-Path-Sidney-I-Resnick/dp/0817684085

[2]

Measures, Integrals and Martingales

Rene' Schilling

https://www.amazon.it/gp/product/1316620247/ref=ox_sc_saved_title_4?smid=A11IL2PNWYJU7H&psc=1

 

Metodi didattici

Traditional black-board based classes and every lecture will be made available through online platforms.

Modalità di verifica e valutazione dell'apprendimento

Written ( and optional oral ) exams. The written exam is articulated in a series of 6 exercises each with a maximum grade of 5 points. Every exercise attains to elements of the syllabus and the relevant Bibliography covered during the course lectures or otherwise hinted at during classes. The (optional ) oral exam may cover the same range of topics as well, while keeping into accont the candidate's perfomance in the previously held written test.

Strumenti a supporto della didattica

Supplementary notes may be distributed during the course itself.

Orario di ricevimento

Consulta il sito web di Enrico Bernardi