When Science Meets Art: Stochastic Analysis and Classical Music

In February-March 2014, the HSE launched six new international laboratories following a tender of three-year projects under the supervision of leading international researchers. One of the projects is the Laboratory of Stochastic Analysis and its Applications.

The international laboratory team gathers researchers from different countries and different universities but who are all linked by close research interests. The HSE News Service keeps on interviewing new staff members of the laboratory. This time Alexander Veretennikov, Leading Research Fellow at the Laboratory of Stochastic Analysis and its Applications, agreed to tell us about his work and plans at the HSE as well as about his hobbies.

— What made you decide to work with the HSE?

— Long ago Valentin Konakov and I started discussing our joint scientific interests and possible collaboration. So, when the application for the International Laboratory at the HSE was preparing it was natural that my name was included. I happily supported it, in particular, because of the convenient format for external participants, which should allow me to continue my research and teaching in the UK while offering a solid basis for this collaboration in Moscow.

— What are the laboratory’s aims?

— Broadly speaking, the goal is the development of stochastic analysis and its applications, which is an essential part of modern mathematical culture. Maintaining its level requires considerable efforts including organisational ones. There are three particular directions in our plans  connected to diffusion processes, which is a stochastic counterpart of the famous heat equation in partial differential equations. We aim to continue and intensify our collaboration between the team members and also to contribute to teaching Master's and PhD students at HSE to attract a new generation of future researchers to this important area. In teaching we are planning to offer several mini courses suitable for Master's and PhD students and also to publish some lecture notes.

 When the application for the International Laboratory at the HSE was preparing it was natural that my name was included. I happily supported it, in particular, because of the convenient format for external participants, which should allow me to continue my research and teaching in the UK while offering a solid basis for this collaboration in Moscow.

— What are your research interests and what are you working on at the moment?

— There are several problems and topics in my current research programme, most of them about stability, approximations and convergence rates for various classes of stochastic processes such as Markov diffusions, diffusions with switching, queueing models, filtering models, et al. One more important part of my research concerns partial differential equations of second order such as Poisson equations. For some models such basic questions like existence and uniqueness are yet to be established; for others the main goal is to study various asymptotic properties. These are the themes I am currently working on, many of them jointly with my co-authors.

— Can you tell us about the courses you are going to teach: Introduction to the theory of Ito SDE and  Introduction to the theory of Pardoux-Peng backward SDEs. Who are they for? What do you need to know to take the course?

— The first course is, in fact, the core of modern stochastic analysis, which is the background of our Laboratory. If any student wishes to join our activities, he has to be educated in this subject. In this way, after attending the course (and hopefully after having passed an exam) such a student will learn the ‘language’ of our research, the terminology which we use while thinking. The student is supposed to follow one or more further courses to start his own research in this field. The compulsory prerequisites for the course comprise basic probability including limit theorems, Lebesgue integration and elements of Markov processes; the knowledge of some elements of function analysis and elliptic and parabolic differential equations of the second order would be desirable. In my mind, such a course should be compulsory for Probability Departments at top universities (in most of them this is realised). I expect that this module may be scheduled for September 2014.

The second topic is more advanced and is not often taught at universities (in Russia I am aware of just one excellent exception in Saint Petersburg); it was chosen among several further chapters in this area and is actually close to some of my own research plans. This subject relates to so called quasi-linear partial differential equations and is also linked to certain issues of control theory. This second course would require more preliminary knowledge. However, attending my first course and, more importantly, successfully passing an exam, should be a sufficient prerequisite. Similar topics were suggested by Professors Menozzi and Belomestnywith whom we have to discuss how to share the subject efficiently.

 My hobbies outside mathematics are classical music and cross-country skiing. I have to say that many of the best research centres in Europe are happy to provide a piano and other musical instruments free of charge to their participants.

— Apart from mathematics, what else interests you?

— My hobbies outside mathematics are classical music and cross-country skiing. Earlier I did kayaking and white water rafting. In skiing I participated thrice in the Tartu skiing marathon. In music I am an ‘educated amateur’ after years at a specialist Music School followed by further years at the Piano Class of Moscow State University and at a Pedagogical Practice Department of the Gnessin Russian Academy of Music. I have to say that many of the best conference/research centres in Europe are happy to provide a piano and other musical instruments free of charge to their participants, for example, Warwick University (UK) and Oberwolfach Research Institute for Mathematics (Germany). This is greatly appreciated by top researchers.