- Docente: Cristina Di Girolami
- Credits: 6
- SSD: MATH-03/B
- Language: Italian
- Teaching Mode: In-person learning (entirely or partially)
- Campus: Bologna
- Corso: Second cycle degree programme (LM) in Mathematics (cod. 6730)
Learning outcomes
At the end of the course, the student will know Ito's stochastic calculus, the fundamentals of stochastic differential equations theory, and its connections to the theory of elliptic and parabolic partial differential equations. The student will be able to independently study both pure and applied mathematical disciplines that require knowledge of stochastic analysis tools.
Course contents
This course presents the theory of stochastic differential equations (SDEs) and their connections with elliptic and parabolic partial differential equations (PDEs).
In particular, the following topics will be covered:
- Review of stochastic calculus: Itô's formula and the theory of stochastic integration
- Stochastic differential equations: existence of strong solutions, uniqueness in law, Markov property, L p L^p Lp estimates, and dependence on initial conditions
- Feynman–Kac formula: connections between stochastic differential equations and the theory of partial differential equations
- Martingale representation theorem
- Girsanov's theorem
- Fundamentals of stochastic control
- Poisson process and compound Poisson process
- Quadratic variation for jump processes
- One-dimensional stochastic integration and stochastic calculus for jump processes
- Extension to the multidimensional case
- Extension to stochastic calculus with semimartingales (overview)
Readings/Bibliography
- Paolo Baldi, Equazioni differenziali stocastiche ed applicazioni, Pitagora Editrice, Bologna 2000.
- Shreve S. E..Stochastic Calculus for Finance 2, Springer, Chapter 11.
Teaching methods
Frontal course
Assessment methods
Oral exam
Teaching tools
Virtuale
Office hours
See the website of Cristina Di Girolami