Advanced Topics in Stochastics Calculus

A.Y. 2020/2021
Overall hours
Learning objectives
The goal of the course is to explore stochastic calculus in depth, by extending the study of stochastic integration from Brownian motion to continuous local martingales. Moreover, based on such an extension, we will turn to other important topics such as: generalised Girsanov theorem, local times and Tanaka's formula as an extension of Ito's formula, weak solutions of stochastic differential equations and Stroock-Varadhan martingale problem, propagation of chaos for mean-field particle systems.
Expected learning outcomes
Detailed knowledge of stochastic calculus for continuous semimartingales (even in a non-regular context) with an introduction to limit theorems for mean-field particle systems.
Course syllabus and organization

Single session

Lesson period
First semester
The lectures will be delivered synchronously through MS Teams, according to the university timetable. They will be recorded and made available on the platform. All the information on the course and on how to have access to the lectures in MS Teams will be provided in the course webpage in ARIEL. It is strongly recommended to check ARIEL regularly.

The program and the course material don't change.

The exams and the assessment criteria don't change, the oral exams will be done in class or through MS Teams depending on the university rules at the time of each specific exam.
Course syllabus
Part I. Review of continuous martingale theory and finite variation processes.

Part II. Stochastic integral with respect to continuous semimartingales, Ito's formula, Dumbis-Dubins-Schwarz theorem on time-changes, generalized Girsanov theorem.

Part III. Ito-Tanaka-Meyer formula, local times of continuous semimartingales, occupation density formula.

Part IV. Weak solutions of stochastic differential equations (SDE), Stroock-Varadhan martingale problem.

Part V. Introduction to mean-field models: nonlinear SDE of McKean-Vlasov type, propagation of chaos.

Part VI. (If time allows) Introduction to large deviations theory.
Prerequisites for admission
1) Advanced notions on probability.
2) Stochastic integral with respect to Brownian motion.
3) Stochastic differential equations (strong solutions).
Teaching methods
Taught lectures
Teaching Resources
1) O. Kallenberg: "Foundations of Modern Probability", Springer, 2002.
2) I. Karatzas, S. Shreve: "Brownian motion and stochastic calculus", Springer, 1998.
3) J.-F. Le Gall: "Brownian Motion, Martingales and Stochastic Calculus", Springer, 2016.
4) D. Revuz, M. Yor: "Brownian Motion and Continuous Martingales", Springer, 1999.
Other references will be given during the course.
Assessment methods and Criteria
Oral exam on the course material. During the exam the student will be asked to describe some of the results of the course in order to assess the knowledge and understanding of the course topics as well as the ability to apply them.
MAT/06 - PROBABILITY AND STATISTICS - University credits: 6
Practicals: 12 hours
Lessons: 35 hours
Professor: Campi Luciano