Geometry of Stochastic Differential Equations

The project is concerned with fine properties of Solutions of Stochastic Differential Equations. On the one hand we try to work out conditions on qualitative behavior of these processes like "Where do they go?", "Do they leave certain sets invariant?". We shall calculate the relevant conditions for Levy-driven Stochastic Differential Equations, a class often applied in Financial Mathematics. On the other hand we try to draw quantitative conclusions about these processes by certain approximation procedures. In mathematical Finance, but also in many other areas of applied mathematics, it is of importance that the approximating processes share some of the qualitative properties with the approximated process. Cubature Formulas are methods to approximate the given process in precisely such a manner. These algorithms are in principle deterministic and also feasable for high dimensions of the modelling space. We try to set up Cubature algorithms in various general cases and we try to make these algorithms feasable by certain recombination arguments. The project is an example of fundamental research motivated in a fruitful way by applications and of applied questions solved by modern pure mathematics.

The project is concerned with fine properties of Solutions of Stochastic Differential Equations. On the one hand we try to work out conditions on qualitative behavior of these processes like "Where do they go?", "Do they leave certain sets invariant?". We shall calculate the relevant conditions for Levy-driven Stochastic Differential Equations, a class often applied in Financial Mathematics. On the other hand we try to draw quantitative conclusions about these processes by certain approximation procedures. In mathematical Finance, but also in many other areas of applied mathematics, it is of importance that the approximating processes share some of the qualitative properties with the approximated process. Cubature Formulas are methods to approximate the given process in precisely such a manner. These algorithms are in principle deterministic and also feasable for high dimensions of the modelling space. We try to set up Cubature algorithms in various general cases and we try to make these algorithms feasable by certain recombination arguments. The project is an example of fundamental research motivated in a fruitful way by applications and of applied questions solved by modern pure mathematics.