This project is concemed with the solution of stochastic differential equations (SDEs). These are mathematical
models for dynamic processes driven by random forcing. A solution is a description of the process in terms of
random quantities, whose probability distributions have to be found. Since these problems are usually too
complicated to be solved exactly, approximations are calculated with the aid of computers. There are two classical
solution strategies: Monte Carlo methods are based an computing large enough numbers of possible experimental
outcomes (so called "paths"), such that the probability distribution of the solution can be estimated. On the other
hand, the fact can be used that ihe desired probability densities are solutions of deterministic problems, usually
partial differential equations (PDEs). For this class of problems a highly developped theory including
approximative solution strategies exist.
The motivation for this project came from a recent work an the approximation of stochastic differential equations
driven by a Wiener (i.e. Gaussian) process. The idea is to use carefully chosen deterministic paths instead of the
random paths in Monte Carlo methods. The choice guarantees that the expectation values of random quantities
derived from the solution of the SDE can be computed with a certain accuracy.
The aim of the project is to develop this idea into a fall fledged numerical approximation procedure and to extend it
to other situations, e.g. when the driving force is a jump process. The project is "interdisciplinary" within
mathematics in the Sense that the expertise of the two main proponents in stochastic processes an the one hand and
in PDEs an the other hand will be combined. As an application, a simulation program for models from
Mathematical Biology describing chemotaxis of certain types of cells (e.g. amobae or leukozytes) will be produced.