The stochastic analysis group has a long and strong tradition in the development of stochastic calculus, including the theory of White Noise and Malliavin Calculus. Of great current interest is to extend such calculus to non-semimartingale processes. The key problems are integration and differentiation based on these processes. Non-semimartingale processes are of interest in many applied areas like finance, turbulence, weather and energy, and we will focus on aspects such as modeling, analysis and simulation.
The group wants to continue to perform high-class research in the field of stochastic partial differential equations (SPDE) and infinite-dimensional stochastic analysis. The activity will be directed towards analysis, modeling and simulation of infinite dimensional processes. Infinitedimensional stochastic analysis is the natural framework for modeling fixed-income, weather and energy markets. Further, also in insurance mathematics infinite-dimensional analysis and SPDEs have been successfully applied to model mortality rates, for instance. The group will continue in this line of research, with more emphasis on the applications of this theory to these markets. For example, issues in hedging and pricing of derivatives are central. It will be natural to collaborate closely with personnel from the PDE-group and from insurance mathematics at the institute.
Stochastic control theory is one of the classical areas of interest for the research group. The area is very challenging, and we want to maintain our interest in this area. Problems we focus on are optimal investment decisions based on various risk measures and access to information. Such seemingly applied questions lead to interesting theoretical problems involving the development of new theory for enlargements of filtrations, analysis of backward stochastic differential equations, and new developments in maximum principles and integro-differential equations. Furthermore, stochastic control problems arising from systems of infinite-dimensional stochastic processes are new areas we would like to enter. But also more empirical issues like simulation and robustness of these problems are of interest, where we want to strengthen the collaboration with the group on insurance mathematics. Finally, a new direction of our group is to collaborate with personnel in the geometry group on combinatorial optimization applied to financial problems.