Stochastic calculus

The course is in Italian. The original title is analisi stocastica.

This is a standard course on the subject. The main arguments are Brownian motion, martingales, Ito integration and introduction to stochastic differential equations.

This is the only course in advanced probability of the degree in mathematics, so from that point of view it must be (and truly is) self-contained. Nevertheless, since some topics are quite advanced, a thorough understanding of probability theory is an expected prerequisite.


Course goals for Students

Gain a good theoretical understanding of stochastic processes and the ability to study simple stochastic differential equations in a qualitative and quantitative way, both in the field of pure research and in industrial applications (for example in finance and in the modeling of noisy systems).


Measure spaces, probability spaces, Borel-Cantelli lemmas, random variables, mathematical expectation, modes of convergence for random variables, Lp spaces.

Lecture topics overview

In the first part of the course we introduce continuous-time stochastic processes and we deal with the new issues arising from this object. In particular, we develop the tools needed for the study of stochastic processes and we show the existence of the Brownian motion. Second part is devoted to the construction of the stochastic integral and to the study of its properties, in particular through martingales. In the third part we give a short introduction to stochastic differential equations.

Recommended readings

  1. Francesco Caravenna - Moto browniano e analisi stocastica (pdf - 155 pages - 3 Mb)
  2. Paolo Baldi - Equazioni differenziali stocastiche e applicazioni
  3. David Williams - Probability with Martingales
  4. Daniel Revuz, Marc Yor - Continuous Martingales and Brownian Motion
  5. Ioannis Karatzas, Steven E. Shreve - Brownian Motion and Stochastic Calculus
  6. Bernt Øksendal - Stochastic Differential Equations: An Introduction with Applications

Assessment methods and criteria


The oral examination consists of three parts. In the first part the student will solve a complex problem assigned some days before by the teacher. In the second part he will be given one or two simple exercises. In the last part he will be asked to state and prove one of the main results of the course.

To pass the exam the student should master the mathematical language and formalism. He must know the mathematical objects and the theoretical results of the course and he should be able to use them with ease. He should also be able to prove theorems by himself.

Teaching methods

Traditional classes (48 hours). Arguments are presented in a formal way, with proofs for most statements. Much stress is given to the motivations and we include some examples of applications. There are no exercise sessions scheduled, but homework is regularly assigned during lessons and students are encouraged to do it at home and possibly ask for solutions during the teacher office hours.

Materials and links


The course was held in the first semester, from October, 9th 2012 to February, 6th 2013.


The course was held in the first semester, from October, 10th 2013 to January, 17th 2014.


The course was held in the second semester, from February, 29th 2016 to June, 8th 2016.