Complex Stochastic Models

Real world phenomena such as physical processes in space, atmosphere and ocean leading to weather and climate, or developments in economics determining prices in finance and risk premia in insurance, can be adequately and efficiently described only by incorporating randomness. Stochastic modelling usually starts by describing systems of real world objects and their dynamics on different spatial and temporal scales and uses different mathematical approaches and techniques:

• microscopic modelling by interacting systems of particles or individual agents with discrete Markovian dynamics in random media,
• mesoscopic modelling by randomly perturbed dynamical systems, stochastic (partial) differential equations, backward equations of stochastic control theory and stochastic delay equations.

For instance, price formation of financial markets starts in a microscopic setting by systems of agents who determine by their combined actions in discrete purchase and sales orders the dynamics of the order book. Limit theorems provide the meso-scopic models of price processes, given for instance by semi-martingales. In a similar way, statistical physics and thermodynamics for particle systems in ocean and atmosphere produces systems of coupled differential equations on different time scales from which stochastic partial differential equations governing the combined dynamics of climate and weather originate.

Generally, the same system can be modelled on different scales, and it is important to study the interaction of different modelling  approaches for instance through limit theorems. Starting on the microscopic level, the analysis of their long-time or average behavior and large deviations give rise to models on the meso-scopic level.

Often, reasonable stochastic models are hard to deduce from microscopic principles. For instance, it is not evident which meso-scopic dynamical model fits best with paleo-climatic data showing the temperature development over a period in earth's history, or high frequency data in finance. In this situation techniques of statistical inference in time series analysis can be instrumental for model selection and calibration.

The agenda of the chair in applied stochastics concerning research, teaching and doctoral training has elements from all levels of stochastic modelling. Its focus is on stochastic analysis and dynamics for meso-scopic models of systems from climate dynamics, financial and insurance mathematics. The main mathematical techniques our work requires are

• dynamical systems perturbed by random noise and their description by stochastic (partial) differential equations; meta-stability under different sources of noise; stochastic resonance; stability and (local) Lyapunov exponents;
• control theoretic methods for the alternative transfer of insurance risk such as backward stochastic differential equations; stochastic calculus of variations or Malliavin calculus; correlation of exposure to risk for hedging concepts;
• statistical inference, model and parameter selection for time series.