OVERVIEW

The course contains an introduction to stochastic calculus and martingale theory that naturally attends in applications in economics.

The teaching contributes to the achievement of Sustainable Development Goals 4 and 5 of the UN 2030 Agenda.

AIMS AND CONTENT

LEARNING OUTCOMES

Introduction to Stochastic Calculus and Martingale theory. Applications in mathematical finance.

AIMS AND LEARNING OUTCOMES

Upon completion of the course, the student will know the fundamentals of the theory of stochastic processes in discrete and continuous time, and will learn the mathematical foundations of Itô stochastic calculus by seeing some of its applications to the financial world. He/she will also be able to handle advanced tools of probability and the theory of stochastic processes.

PREREQUISITES

Program of the course "Probability.

TEACHING METHODS

Lectures + laboratory

SYLLABUS/CONTENT

Conditional hope/filtrations and stopping times/discrete-time martingale (Doob's inequalities, convergence results; Doob-Meyer decomposition)/continuous-time martingale (Continuous or continuous right-hand trajectory versions with limits from the left. Kolmogorov's theorem. Continuous-time martingales and their properties. Martingales closed by an integrable or integrable square random variable)

Brownian motion (scale change invariance property, strong Markov property, reflection principle, law of maximum, level crossing times, geometric Brownian motion, multidimensional Brownian motion, recurrence and transience. Brownian motion with drift. Ornstein-Uhlenbeck process, Bessel process, Brownian bridge.)/ Stochastic integration/Simulation MB/ Applications to finance.

RECOMMENDED READING/BIBLIOGRAPHY

A. Pascucci, PDE and Martingale methods in Option Pricing, Bocconi & Springer Series (2010)

P. Baldi, S. Shreve, Stochastic Calculus and Finance

P. Baldi, Equazioni differenziali stocastiche e applicazioni, Bologna, Pitagora (2000)

Mörters, Peres, Brownian Motion

F. Caravenna, Moto browniano e analisi stocastica

TEACHERS AND EXAM BOARD

Exam Board

VERONICA UMANITA' (President)

DAMIANO POLETTI

EMANUELA SASSO (President Substitute)

LESSONS

LESSONS START

According to the academic calendar approved by the Course Council.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Oral examination.

Students with certified DSA, disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination arrangements that, while respecting the teaching objectives, take into account individual learning patterns and provide suitable compensatory tools. In order to request exam accommodations, the instructions detailed on Aulaweb

https://2023.aulaweb.unige.it/course/view.php?id=12490#section-3

should be followed. In particular, accommodations should be requested significantly in advance (at least 10 days) of the exam date by writing to the teacher with a copy of the School Referring Teacher and the appropriate Office (see instructions).

ASSESSMENT METHODS

Verification of learning is through oral examination only and will focus on topics covered in class. The student will be expected to show correctness of language and mathematical formalism, to have a deep knowledge of mathematical objects and results of the course, and be able to use them naturally.

Exam schedule