B0324 - EQUAZIONI DIFFERENZIALI STOCASTICHE I

Learning outcomes

At the end of the course, the student deals stochastic calculus according to Itô, the foundations of the theory of stochastic differential equations and the links with the theory of partial differential equations of the elliptic and parabolic type. He is able to independently conduct the study of pure and applied mathematical disciplines that require knowledge of stochastic analysis tools.

Course contents

The main theory of stochastic differential equations and links with elliptic and parabolic partial differential equations. In particular, the following topics will be covered:

• Remind of stochastic calculus: Ito formula, theory of stochastic integration
• Stochastic differential equations: existence of strong solutions, uniqueness in law, Markov properties, estimates Lp and dependence on initial data
• Feynman-Kac formula: link between stochastic differential equations and the theory of partial differential equations
• Martingale representation theorem
• Girsanov theorem
• Weak solutions: uniqueness in law, Tanaka example and Yamada-Watanabe theorem
• Martingale problem
• Regularization by noise: Zvonkin theorems, weak good position for equations with measurable drift, and strong for equations with Holderian drift
• Stochastic partial differential equations: introduction to the problem of stochastic filtering

Readings/Bibliography

Paolo Baldi, Stochastic Calculus: An Introduction Through Theory and Exercises, Springer.

Course notes with references to the professor's Pascucci notes https://1drv.ms/u/s!AqFHqfUowiJlj-c2YzqLZUFDWnIQXA?e=XM2Bw3

Teaching methods

Frontal lessons and exercises.

Assessment methods

The exam consists of an oral test with exercises and theory.

Teaching tools

Platform on virtuale.unibo.it where it is available: notes of the lectures, exercises and all the informations for the course.

Office hours

See the website of Cristina Di Girolami

See the website of Antonello Pesce