Risorse bibliografiche
Risorsa bibliografica obbligatoria
Risorsa bibliografica facoltativa
Scheda Riassuntiva
Anno Accademico 2018/2019
Scuola Scuola di Ingegneria Industriale e dell'Informazione
Docente Confortola Fulvia
Cfu 8.00 Tipo insegnamento Monodisciplinare

Corso di Studi Codice Piano di Studio preventivamente approvato Da (compreso) A (escluso) Insegnamento

Obiettivi dell'insegnamento

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.

Risultati di apprendimento attesi

Knowledge and understanding

Lectures and exercise sessions will allow students to

- know the fundamental elements of stochastic processes (Brownian motion, martingales and Markov processes), stochastic calculus and stochastic differential equations;

- understand the applications: there are very many situations (in finance, telecommunications, control, ...) where stochastic processes, and in particular diffusions, are a natural model;

 - understand the connections with other branches of pure mathematics, in particular with partial differential equations.



Ability in applying knowledge and understanding

Lectures and exercise sessions will allow students to

- manipulate the basic notions of stochastic calculus acquiring elements of the theory and ability to work with them;

- have an adequate knowledge of the techniques to formalize and to model evolutions with noise in different contexts of science and technology;

- learn the constructive mathematical proofs of some key result of the theory;

- acquire the ability to present in a clear and precise way the models and the results of the theory.

Argomenti trattati

1. Elements of Probability. Probability spaces, random variables. Variance, covariance, probability distribution, density. Independence. Random vectors. Convergence of random variables. Characteristic functions. Gaussian laws. Measurability theorems.

2. Stochastic processes. Filtrations. Trajectories. Equivalent processes, modifications, indistinguishable processes. Finite-dimensional laws and Existence Kolmogorov’s theorem. Kolmogorov’s continuity theorem. Stopping times.

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

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.


The only prerequisite is a first course in Probability or equivalent and general notions in Measure and Integration theory are required.

Modalità di valutazione

The final exam consists in

- a preliminary written test, during the exam sessions;

- an oral test on the full programme (maximum score 30/30 cum Laude) or on a reduced programme (maximum score 28/30).

The candidate can access the oral test if he/she totalizes at least 18/30 in the written test. The written and the oral test have to be passed in the same exam session.


The convocation to the oral test will be communicated by an announcement on the Beep page of the teaching.


The written test usually consists of 3 exercises, each one containing 7 – 10 questions, on

- stochastic processes, and in particular Brownian Motion, and their properties;

- conditioning, martingales and their applications in the investigation of stochastic processes;

- stochastic calculus: stochastic integral and Ito’s formula;

- stochastic differential equations: existence and uniqueness of the solution and its properties.


During the written test it is not allowed neither to consult documents or archives nor to communicate with other people.  In particular it is not allowed to keep on the mobile phone or some other device allowing to communicate, or to use internet, to consult documents.


The candidate will be asked to solve problems with computations and to justify rigorously all passages. He/she will be required to possess the elements of the theory and to be able to manipulate the theoretical notions.


The evaluation of the written test will take into account the clarity in the exposition of the used procedures, the skill to motivate them, the accuracy in the computations, the familiarity with mathematical language. Each written test will be evaluated with a score between 0 and 30. The candidate will pass the written test if he/she obtains a score bigger than or equal to 18.


The oral exam consists in questioning on topics of the course. The candidate can choose to do the oral test on the full programme (in this way he/she  has the possibility to obtain as maximum score 30/30 cum Laude), or on a reduced programme, but in this case the maximum score will be 28/30. The reduced programme contains the same topics of the full programme, with the same definitions, the same examples and the statements of the same theorems, but contains less proofs. During the lessons for each theorem it will be specified if the related proof belongs to the full programme only or also to the reduced programme. At the end of the semester will be published on the BeeP page of the teaching a detailed list of the proofs of the full programme only.


The candidate will be asked to introduce the fundamental notions of the stochastic calculus, to summarize the main properties of mathematical tools of stochastic calculus, to state and to prove the main theoretical results. Moreover, the candidate is supposed to be able to present rigorously with an adequate and specialized dictionary and mathematical formalism the results and the applications and to possess skill at making connections between the teaching topics.


At the end of the oral test it will be formulated a score, taking into account all tests: if the evaluation is positive (bigger than or equal to 18) and it will be accepted by the candidate, the score will be verbalized. Otherwise the student will have to take again the written and the oral test, during the following exam sessions.


More information can be found in the Sillabo of the teaching published on BeeP, https://beep.metid.polimi.it/.



Risorsa bibliografica facoltativaP. Baldi, Stochastic Differential Equations

Draft of a forthcoming book, courtesy of the Author.

Risorsa bibliografica facoltativaP. Baldi, Equazioni differenziali stocastiche e applicazioni, Editore: Pitagora Editrice, Bologna, Anno edizione: 2000, ISBN: 88-371-1211-4
Risorsa bibliografica facoltativaF. Caravenna, Moto Browniano e Analisi Stocastica, Anno edizione: 2011 http://www.matapp.unimib.it/~fcaraven/download/other/dispense-3.3.pdf
Risorsa bibliografica facoltativaB. Oksendal, Stochastic Differential Equations, Editore: Springer, Anno edizione: 2000, ISBN: 3540637206
Risorsa bibliografica facoltativaS.E. Shreve, Stochastic calculus for finance - Continuous-time models (Vol 2), Editore: Springer, Collana Springer finance, Anno edizione: 2004, ISBN: 0387401016

Forme didattiche
Tipo Forma Didattica Ore di attività svolte in aula
Ore di studio autonome
Laboratorio Informatico
Laboratorio Sperimentale
Laboratorio Di Progetto
Totale 80:00 120:00

Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua Inglese
Disponibilità di materiale didattico/slides in lingua inglese
Disponibilità di libri di testo/bibliografia in lingua inglese
Possibilità di sostenere l'esame in lingua inglese
Disponibilità di supporto didattico in lingua inglese
schedaincarico v. 1.6.5 / 1.6.5
Area Servizi ICT