I algebraic geometry

Research mission and objectives:

The objective of the research is to study various properties of stochastic processes, and the use of them in modeling of risk. The study on risk, its modeling and even control under risk is one of the most important challenges of quantitative methods in economy. The following aspects of quantitative methods in economy. The topic of the research is closely related with real applications. Industry, in particular financial and insurance industry is interested in special departments which are monitoring risk.

Risk is an inevitable part of all human activity. It appears almost in everywhere: in construction works, production, investments, finance and insurance and in our everyday life. The notion of “risk” carries connotations of chaos, the unexpected and undesired behaviour of an observed phenomenon. It is difficult to anticipate how a chaotic model will behave, since- as the time itself indicates- such a model does not have predictable dynamics. But while it is hard to say how a chaotic process will behave at any specific moment, things are quite different if we take a longer-term perspective, chiefly looking at the mean value of certain functions that hinge on the process. For example, the investors of a bank or insurance company are much more interested in the overall market situation than in individual transactions they might stand to lose on. The point is to be sure they “come out on top” in the appropriate long-term perspective, regardless of short-term fluctuations. Calculating risk requires a certain language (or mathematical model) to describe the observed phenomena. In this project we shall concentrate ourselves on the mathematical aspects of risk with potential applications to finance and insurance. We are also interested in proper pricing of financial derivatives which should take into account various aspects of risk.

To study risk it is important to investigate various properties of stochastic processes. The following specific aspects of stochastic processes in particular will be studied: solutions to stochastic differential and partial differential equations, possibly with Levy noise with applications to mathematics of finance and insurance. We will be interested in regularity (strong Feller property, existence of densities and existence (uniqueness) of invariant measures) of solutions to such equations.

Another objective is to study ergodic properties of various Markov processes: solutions to stochastic differential and partial differential equations, filtering processes and also to use these properties to control models with ergodic type criteria.

Further topic is a comprehensive analysis of  risk measures and financial markets under various kinds of risk. In particular an illiquid markets and markets with frictions.