In this project we apply modern stochastic methods to study a number of analytical and applied problems. Many phenomena in science and everyday life exhibit inherent uncertainty. We commonly use stochastic models to describe such behaviour and rely on stochastic methods to gain insights, make forecasts or inference about them. Various stochastic methods have appeared over the years in biology, medicine, biomedicine, geology, climatology, social sciences, finance and insurance, and many other scientific fields. The area of mathematics that lies in the background is probability theory, in particular, theory of stochastic processes. It is a rich mathematical theory with proven potential to solve not only applied problems, but also some quite theoretical problems coming from other areas of mathematics.
The problems that we plan to study can be broadly divided into four groups:
This choice of problems is influenced by importance in the current international research as well as the background and expertise of members of our team. The goal of the project is to advance understanding of the role of randomness in each of these four settings. The unifying feature behind our approach to these problems is common stochastic methods in their analysis. These methods include, but are not limited to, martingale methods, point processes and random measure methods, potential-theoretic methods (both probabilistic and analytic), Markov processes methods, methods from the Dirichlet form theory, stochastic integrals, diffusion processes, branching processes and measure-valued processes methods, time-series methods, wavelets and methods for statistical inference.