B029716 - STOCHASTIC PROCESSES

English

For Mathematicians: Brief introduction to martingales, Brownian motion, stochastic integrals, Ito calculus and stochastic differential equations. First notions of Cryptography and Blockchain technology. [For Computer scientists: NP problems (factorization of prime numbers, elliptic curve, quantum and probabilistic algorithms) Case study: financial area (cryptocurrencies, options, pricing of derivative securities, interest rate curve, portfolio management)

B. Oksendal - Stochastic Differential Equations. An Introduction with Applications
I. Karatzas, S. Shreve - Brownian Motion and Stochastic Calculus

Paul Wilmott - Sam Howison - Jeff Dewynne: The Mathematics of Financial Derivatives- Cambridge University Press

Appunti forniti sulla piattaforma Moodle

The course aims to provide all students with basic knowledge of stochastic calculus and its applications to financial mathematics. Moreover, students will be provided with basic knowledge and understanding of blockchain technology. Mathematics students will be provided with in-depth knowledge of Stochastic Calculus, while those of Informatics will have in-depth knowledge of the algorithms used in cryptography. One of the aims is to let the students develop basic technical skills, and critical thinking, needed when modelling and solving mathematical and computer science problems in different settings. Special attention will be paid to help the students to develop communication skills necessary for teamwork. The course covers topics and provides learning skills that are needed, or strongly suggested, to pursue a degree in applied mathematics\computer science.

Courses to be used as requirements (required and/or recommended):
Required courses: Probability, Mathematical Analysis I and II
Recommended courses: Algorithms and data structures

CFU: 9
Total hours of the course: 220
Hours reserved to private study and other indivual formative activities: 148
Contact hours for: Lectures (hours): 72

Lectures: Presentation of the theory described in the course program, with teacher-student direct interaction, to ensure a full understanding of the subject.


Moodle learning platform: online teacher-student interaction and posting of additional notes.

Remark: The suggested reading includes supplementary material that may be useful for further personal studies in mathematics or in any scientific subject.

Office hours:
prof. Vespri: 4 hours a week. Appointment to be taken via e-mail. In general, prof. Vespri prefers Monday and Tuesday from 1.30 p.m. to 3.30 p.m.

Contacts of prof. Vespri:
Dipartimento di Matematica e Informatica "Ulisse Dini"
Viale Morgagni, 67/a
50134 FIRENZE
Tel: 055 2751405
Email: vespri@math.unifi.it

prof. Avena: Appointment to be taken via e-mail. Precise time/day to be discussed with students.

Contacts of prof. Avena:
Dipartimento di Matematica e Informatica "Ulisse Dini"
Viale Morgagni, 65
50134 FIRENZE
Tel: 055 2751462
Email: luca.avena@unifi.it

Oral examination: A number of questions are posed.

Math students: there will be both a written exam (exercises, theorem statements and proofs) and an optional oral exam (theorem statements and proofs) on the stochastic calculus portion of the course. The oral examination is designed to evaluate the degree of understanding of the theory presented in the course. In the assessment, special attention is paid to communication skills, critical thinking and appropriate use of scientific language.

Topics:

1.
For mathematicians:
- Problems motivating the study of SDEs
- Brownian motion: existence and properties
- Ito integral: construction and properties
- Ito calculus, Martingale representation theorem
- Stochastic differential equations: weak and strong solutions, existence and uniqueness, solution methods
If time allows:
- Applications to filtering problems
- Diffusion, Kolmogorov equations, Feynman-Kac formula, Girsanov theorem
- Applications to stochastic control

2.
- Financial Markets, Stocks and Bonds, European and Asian options
- Black and Scholes model, Black and Scholes equation, Partial differential equations (outline), American options - problems with obstacle and free boundary (outline),
- Numerical solution of the equations of Black and Scholes (European options), Finite difference method, LSU, SOR, Crank-Nicholso,
Binomial method, Numerical solution of the equations of Black and Scholes (American options)
exotic options, compound options, chooser options, barrier options, Asian options, lookback options, Russian options, Stop loss options, Options with transaction costs
- Pricing of bonds, the yield curve, stochastic interest rate, equation for the pricing of bonds, options on bonds
- Swaps, Floors, Caps, Options on swaps, caps and floors
convertible bonds, convertible bonds with an interest rate stochastic

3.
- Blockchain technology.
- The problem of consent.
- The first generation of cryptocurrencies.
- Blockchain 2.0 and 3.0
- Applications
- Preliminaries of Cryptography .

4.
For Computer Science students:
- Algorithms for the factorization of prime numbers
- Algorithms for solving problems related to elliptic curves
- Quantum algorithms for solving complete NP problems

Quality education and Industry, innovation and infrastructure