93013 - Stochastic Processes

Learning outcomes

Students will acquire new mathematical/probabilistic skills in the field of stochastic processes becoming more self-conscious in applying them to models of financial market and any pricing issues.

Course contents

The course is devoted to the theory of Stochastic Calculus for Brownian Motion. This constitutes the fundamental framework for studying the problems of pricing and hedging of financial instruments. The content can be schematized as follows:


• Brownian motion: definition and basic properties
• Ito integral: definition and key features
• Extension of the Ito integral and examples
• Ito's formula and applications
• Stochastic differential equations: existence, uniqueness and examples
• Numerical methods and simulation of stochastic differential equations
The discussion of all the theoretical tolls and results will be supported by a series of exercises with solutions to help the implementation of the different aspects of the theory to concrete problems.

Readings/Bibliography

Lecture notes made available at the start of the course together with exercises with solutions delivered at the end of each week of lectures. This material will be posted on Virtuale.

Further suggested readings:

• H. H. Kuo, Introduction to Stochastic Integration, Springer, 2006
• B. Øksendal, Stochastic differential equations - VI edition, Springer, 2003
• R. L. Schilling and L. Partzsch, Brownian Motion. An Introduction to Stochastic Processes, De Gruyter, Berlin, 2012

Teaching methods

Regular lectures and tutorials.

Assessment methods

This course is part of the Integrated Course "Stochastic Processes and Econometrics". The assessment for the Stochastic processes' part consists in an open-book, one hour written exam, articulated in a series of 2 exercises each with a maximum grade of 15 points. Every exercise attains to elements of the syllabus covered during the course lectures.The exam aims to test the student's degree of skill in using the techinques developed during the course for the solution of standard exercises.To pass the exam a minimum grade of 18/30 is required.

The final grade for the course "Stochastic Processes and Econometrics" is obtained by taking the arithmetic mean of the grades achieved in the two different parts of the course.

In case of online exam, this will be supported by the softwares Teams, Zoom and EOL (https://eol.unibo.it/)

Teaching tools

Slides, tutorials and exercises with solutions

Office hours

See the website of Alberto Lanconelli