Laurea Magistrale (M.Sc.) in Stochastics and Data Science

Stochastic differential equations

Stochastic differential equations

Academic year 2020/2021

Sommario del corso

• Course objectives
• Results of learning outcomes
• Course delivery
• Learning assessment methods
• Program
• Suggested textbooks and readings
• Notes
• Teaching tools

Course objectives

The course aims to put the student in a position to understand the mathematical formulation of various models of applied sciences and financial mathematics which involve stochastic differential equations. The course uses probabilistic concepts and tools that are developed in the course ``Probability Theory''  and elements of Functional Analysis  (see ``Analysis''); these concepts  are briefly mentioned in the first lectures.   The proofs of the main results of the course are carried out completely. They show important links between Analysis and Probability. To improve the skills of reading and  study  the teacher proposes the reading of some scientific articles.    Together with the course ``Stochastic Processes''  it suggests an approach to the research in stochastic environments. The course also provides basic concepts on  parabolic equations of Kolmogorov type.

A module of the course, included in the overall courseload, will be taught by visiting professor Franziska Kuhn (University of Manchester, UK) on "Some path properties of BM" (cf. International visiting professorsopen_in_newopen_in_new).

Results of learning outcomes

At the end of the course, students will know several important methods to study stochastic models; in particular  they will know the Ito stochastic integral and the related stochastic differential equations. Moreover, they will understand relations between stochastic differential equations and Kolmogorov equations. They will be able to study applications of stochastic differential equations to solve problems in applied sciences

Course delivery

The course is composed of 48 hours of lectures which, for the AY 2020/2021, will be held remotely, either as live streaming or pre-recorded. All lectures will be recorded and made available on Moodle in due time, together with the related slides or notes and other course material.

Some additional activities to favour direct interaction between professors and students may be organised as online meetings and/or meetings in presence, under appropriate conditions of social distancing and compatibly and in compliance with future existing regulations. For meetings in presence, students who are not able to be physically present will have the chance to follow such activities through the online course material

Learning assessment methods

Oral examination. Questions on the program (theorems with some proofs, remarks and examples). 

DURING THE SANITARY EMERGENCY FOR THE DIFFUSION OF COVID THE ASSESSMENT METHODS WILL BE UNCHANGED, BUT ALL PROCEDURES WILL BE CARRIED OUT ONLINE USING WEBEX. STUDENTS ENROLLED TO THE EXAM WILL RECEIVE A LINK FOR THE WEBEX MEETING.

Program

-  Reminder of basic notions  on  measure theory and probability theory. Multidimensional Gaussian distributions.

-  Brownian motion  (its construction by means of Kolmogorov's theorem); the Wiener measure. Global and local path properties of Browian motion

- The Ito stochastic integral  (basic properties; comparison between the stochastic integral and the  Riemann-Stieltjes integral) 

- The Ito formula and its applications 

- Stochastic differential equations (existence and uniqueness theorems)

- Markov property of solutions of stochastic differential equations; connections between  stochastic differential equations and parabolic Kolmogorov equations

- Possible applications of  stochastic differential equations  to Mathematical Finance and Population Dynamics 

- Arnold, L., Stochastic Differential Equations, Theory and Applications, New York. John Wiley & Sons. 1974

- P. Baldi: Stochastic Calculus. An Introduction Through Theory and Exercises.  Springer, 2017

- P. Baldi: Equazioni differenziali stocastiche e applicazioni, Pitagora Ed., Bologna, 2000.

- I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, Second Edition, 1991.

- R. Schilling, L. Partzsch and B. Bottcher. Brownian Motion: An Introduction to Stochastic Processes. De Gruyter.

- Wilmott P., Dewynne J. and Howison S. The mathematics of financial derivatives: a student introduction. Cambridge University Press, 1995.

Class schedule

Note

This course will be delivered at the ESOMAS Department.

• Teaching materials
• Exams and finals
• Go to Moodle
• Visit the forums