- Docente: Cristina Di Girolami
- Credits: 6
- SSD: MAT/06
- Language: Italian
- Moduli: Cristina Di Girolami (Modulo 1) Salvatore Federico (Modulo 2)
- Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
- Campus: Bologna
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Corso:
Second cycle degree programme (LM) in
Mathematics (cod. 6730)
Also valid for Second cycle degree programme (LM) in Mathematics (cod. 5827)
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
Prerequisite: the course in Stochastic Calculus untill Ito's Formula.
The main theory of stochastic differential equations and links with elliptic and parabolic partial differential equations.
Introduction to theory of stochastic calculus in jump processes.
In particular, the following topics will be covered in the first part of the course.
- 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
In the second part of the course
Introduction to Jump processes.
In particular:
- Construction of the Poisson process
- The law of the Poisson process
- Compound Poisson process
- The law of the compound Poisson process
- Stochastic integration with jump-diffusions
- Ito's formula for jump diffusions
- Linear stochastic differential equations and Girsanov's Theorem for jump-diffusionos
- Market models with Poisson process: absence of arbitrage and pricing
- Market models with Poisson process and Brownian motion: absence of arbitrage and incompleteness
Readings/Bibliography
Paolo Baldi, Stochastic Calculus: An Introduction Through Theory and Exercises, Springer.
Shreve S. E..Stochastic Calculus for Finance 2, Springer, Chapter 11.
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 Salvatore Federico