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Scheda Riassuntiva
Anno Accademico 2014/2015
Scuola Scuola di Ingegneria Industriale e dell'Informazione
Insegnamento 095975 - STOCHASTIC DIFFERENTIAL EQUATIONS
Docente Gregoratti Matteo Probo Siro Francesco
Cfu 8.00 Tipo insegnamento Monodisciplinare

Corso di Studi Codice Piano di Studio preventivamente approvato Da (compreso) A (escluso) Insegnamento
Ing Ind - Inf (Mag.)(ord. 270) - MI (403) INGEGNERIA MATEMATICA* AZZZZ095975 - STOCHASTIC DIFFERENTIAL EQUATIONS
Ing Ind - Inf (Mag.)(ord. 270) - MI (487) MATHEMATICAL ENGINEERING - INGEGNERIA MATEMATICA* AZZZZ095975 - STOCHASTIC DIFFERENTIAL EQUATIONS

Programma dettagliato e risultati di apprendimento attesi

 

STOCHASTIC DIFFERENTIAL EQUATIONS

 

 

 

Obiettivi e contenuti del corso

 

1. Introducing Stochastic Calculus and its rules.

 

2. Introducing Stochastic Differential Equations.

 

3. Showing their utility in modelling evolutions with noise in different contexts such as Finance,

 

Engineering, Chemistry, Physics.

 

 

 

 

 

Descrizione degli argomenti trattati

 

1. Elements of Probability. Probability spaces, random variables. Variance, covariance, probability

 

distribution, density. Independence. Random vectors. Convergence of random variables. Charac-

 

teristic functions. Gaussian laws. Measurability theorems.

 

2. Stochastic processes. Filtrations. Trajectories. Equivalent processes, modifications, indistinguish-

 

able processes. Finite-dimensional laws and Existence Kolmogorov’s theorem. Kolmogorov’s conti-

 

nuity theorem. Stopping times.

 

3. Brownian motion. Definition and basic properties. Finite-dimensional distributions. The White

 

4. Conditional probability. Conditional expectations. The augmented Brownian filtration.

 

5. Martingales. Definitions and basic properties of continuous time Martingales.

6. The stochastic integral. Elementary processes. The stochastic integral. Ito Isometry. The stochastic integral as a process. Stopping theorems. Local martingales.

7. Stochastic calculus. Stochastic differential of an Ito process. Ito's Lemma. Girsanov’s Theorem. The martingales of the Brownian filtration.

8. Stochastic Differential Equations. A class of SDE. Definition of solutions. Existence and Uniqueness

theorems for the solution. SDE and Markov processes. Connections between SDE and PDE.

Feynman-Kac formula.

 

Organizzazione del corso e modalità di verifica

The course final exam is made of a preliminary written test, followed by an oral test, whose access is

subject to a passing rate in the written test.

 

 

 


Note Sulla Modalità di valutazione

The course final exam is made of a preliminary written test, followed by an oral test, whose access is

subject to a passing rate in the written test.

 


Bibliografia

Mix Forme Didattiche
Tipo Forma Didattica Ore didattiche
lezione
50.0
esercitazione
30.0
laboratorio informatico
0.0
laboratorio sperimentale
0.0
progetto
0.0
laboratorio di progetto
0.0

Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua Inglese
schedaincarico v. 1.6.1 / 1.6.1
Area Servizi ICT
22/02/2020