The main focus of my research in the past have been stochastic processes which are solutions of stochastic differential equations (short SDEs) driven by so-called Levy processes, in particular stable processes. Levy processes is a natural and a large class of processes occurring in many applications, including Brownian motion, Poisson process and many other “canonical” processes. Brownian motion is the only one among Levy processes which has continuous sample paths while all other Levy processes are so-called “jump processes”. The interest to models involving jump Levy processes has increased recently since they do provide a more accurate description to many concrete models arising in practice.
Among solutions of stochastic differential equations one distinguishes so-called “strong solutions” and “weak solutions”. The theory of strong solutions goes back to pioneering works of Kiyosi Ito who laid the foundations of stochastic analysis in 1940th. Weak solutions were introduced first by A. V. Skorokhod in 1956.
I am interested mainly in questions of existence, uniqueness and properties of weak solutions of SDEs with measurable coefficients driven by Levy processes. The theory of weak solutions of SDEs with only measurable coefficients is far from being complete as measurable coefficients provide the biggest class of functions for which such solutions can be constructed. The first result for an SDE with only measurable coefficients was obtained by N. V. Krylov in 1980 for the Brownian motion. He was able to construct solutions in that case using some integral estimates he was also first to derive. Such estimates are now referred to as Krylov type estimates. There is much less known for SDEs with only measurable coefficients driven by processes different from a Brownian motion.
Among some of our contributions to the theory of weak solutions of SDEs driven by jump Levy processes are the derivations of various forms of integral estimates of Krylov type for purely jump SDEs having the presence of the drift term. As an application of those estimates, weak solutions for new classes of SDEs were constructed. The results obtained are also relying on some facts from the optimal control theory for stochastic process and corresponding Bellman equations as well on some analytic a priori estimates for solutions of related parabolic integro-differential equations.
I am also interested in some applications of stochastic differential equations including the fields of engineering and finance.
A note to students: if you have passion for mathematics and are interested in doing a Master thesis with a significant mathematical component, you can talk to me about a possible research topic.