Transport approach to mimicking processes

When working with stochastic processes, one can often make statements about the expected values at certain points in time, but the probabilistic behavior over longer time periods is much more difficult to understand. One example that is particularly important for applications concerns the future prices of stocks or securities: Here it is possible to derive statements about the distribution on individual days from derivative prices, but with what probability the price process will follow a particular path is a much more difficult question. An important challenge in stochastics is therefore to construct stochastic processes as simply and naturally as possible that respect known information about the behavior on particular days (in particular, known "marginal distributions"). Although the first important works on this topic date back to Strassen (1965) and Kellerer (1972), it has not yet been possible to develop an adequate systematic theory to solve this problem. The aim of this research project is to develop such a theory. This is based on new techniques of probabilistic transport theory, which have been developed only in recent years. We expect that the results of the project will be both stimulating for the theory of stochastic processes and directly important for mathematical finance applications.