The Research Program

The research program of the IRTG is structured into three clusters, namely
 I. Stochastic Analysis
 II. Mathematical Physics
 III. Models in Finance and Economics.

However, one of the core ideas of our research program is to connect these clusters and their different focal areas of research, and to develop the understanding of interactions between them.

Below, we provide an overview of the clusters and the interactions between the different clusters.

Cluster I -- Stochastic Analysis

1. Dirichlet Forms and SPDE

2. Functional Inequalities and Heat Kernel Bounds

3. Free and Quantum Probability

Cluster II -- Models in Mathematical Physics

1. Configuration spaces in the continuum

2. Complex networks

3. Applications of Infinite Dimensional Analysis

Cluster III -- Models in Finance and Economics

1. Robust Financial Market Models

2. Equilibria in Incomplete Markets, Pairwise Random Matching, and Aggregation

3. Strategic Issues in Financial Markets

Interactions

Cluster I -- Cluster II:

Stochastic dynamics analyzed in Cluster II, Subproject II.1, can no longer be described by an SPDE, but only via their generators, which in turn constitute a very new interesting class of Kolmogorov operators, generating generalized Dirichlet forms as studied in Cluster I. Furthermore, spectral analysis provided by functional inequalities and heat kernel analysis in Cluster I is vital for investigating e.g. the long time behaviour of the dynamics studied in Cluster II. Finally, various scaling limits, asymptotic distributions and processes in random environment are focal points of research both in Cluster I and Cluster II, from the more theoretical point of view in the first and from the more modelling and applied view point in the latter.

Cluster I -- Cluster III:

BSDE studied in Cluster I are heavily applied in stochastic finance, a main theme in Cluster III. Fractional Brownian motions, studied in Cluster I, because of their heavy tails, are considered to describe noise terms in stochastic dynamics occuring in finance more accurately than geometric Brownian motion. The same is believed to be true for Levy noise driven SPDE. The latter are another focal point in Cluster I.

Cluster II -- Cluster III:

Methods from theoretical and mathematical physics, in particular from percolation theory (phase transitions in economic systems), self-organized criticality (avalanche dynamics in financial markets) and the theory of complex networks (agent based models, robustness, synchronization of economic systems) studied in Cluster II can be useful and natural to model and analyze different economic and financial systems as studied in Cluster III.

Different kinds of spatial stochastic dynamics considered in Cluster II have their origin and applications in economical models and financial mathematics. In particular, a model in which new economic units independently appear in a financial market has been considered by Y.Kondratiev and D.Finkelshtein. It was shown that that such a market can be stabilized through a local competition mechanism without any global regulation.