2 semester


Title of discipline


Functional methods in the theory of differential equations


Course of study



1, 1-31 80 03 Mathematics


Semester of training

(for each semester a separate table)




Number of credits





Name and surname lecturer


Yablonsky Oleg Leonidovich


Objectives of the study of the discipline


     Training of specialists capable of using fundamental mathematical knowledge as a basis for carrying out applied research.

Teaching discipline solves the following tasks:

     familiarization of undergraduates with the methods of stochastic analysis, differential equations and algebras of generalized random processes;

     formation of abilities for master students to independently develop algorithms for solving problems and analyze them;

     develop and use tools, information environments, automated systems;

     use mathematical methods of research in the analysis of modern natural-science, economic, socio-political processes;

     acquisition of abilities independently to expand the knowledge with their further use at the analysis of mathematical models of a wide range of applied problems.

     As a result of studying the academic discipline, the master of the master’s degree must:


–        basic concepts of stochastic differential equations;

classical methods of functional analysis;

–        standard research methods;

be able to:

–        correctly apply the basic methods;

–        solve standard stochastic differential equations;


–        methods of functional analysis;

–        methods of stochastic analysis.




Fundamentals of Probability Theory

Fundamentals of functional analysi



Contents of the discipline

Random processes. Kolmogorov’s theorem. The process of Brownian motion and its properties. Martingales. Filtration and flow of sigma algebras. Stochastic integrals of Ito and Stratonovich and their properties. Ito’s formula. Stochastic differential equations. The concept of a strong and weak solution. Potrajectory uniqueness and uniqueness in distribution. The existence and uniqueness theorem for a solution. Definitions of algebras of generalized random processes. Generalized differential. Equations in differentials in algebras of generalized random processes


Recommended Reading


1. Lazakovich NV, Stashulyonok SP, Yablonsky OL. Probability theory. – Minsk, BSU, 2007.

2. Wentzel A.D. Course of the theory of random processes. – M., “Science”, 1978.

3. Watanabe S., Ikeda N. Stochastic differential equations and diffusion processes. – Moscow: Nauka, 1986.

4. Oxendal B. Stochastic differential equations. – The World, AST, 2003.


Methods of teaching

interactive teaching methods (working in small groups (team), problem training) are organized taking into account the inclusion in the learning process of all students of the group. Joint activity means that each student makes his own individual contribution, in the course of the work there is an exchange of knowledge, ideas, methods of activity. Organized individual, steam and group work. Interactive methods are based on the principles of interaction, activity of trainees, reliance on group experience, mandatory feedback



Language of learning




Conditions (requirements), routine control


Test papers



Appraisal form