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

Stochastic differential equations

Stochastic differential equations

Academic year 2018/2019

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.

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

Lectures (48 hours, 6 CFU) are given in lecture rooms.

Learning assessment methods

Oral examination. Questions on the program (theorems, remarks and examples).  Concerning the proofs we require to know in details 3 important proofs. Such required proofs are given in the folder ``Teaching material'' below. This folder also contains more information on the examination.

Program

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

-  Brownian motion  (its construction by means of Haar functions; regularity properties of trajectories); the Wiener measure. 

- The Doob L^p estimates for martingales with continuous paths.

- 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 

- Lectures notes

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

- 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.

Class schedule

Note

This course will be delivered at the ESOMAS Department.

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