Randomness is ubiquitous in real-life phenomena, for instance on trading floors, in materials undergoing phase transitions, in the genealogy of populations or in terrestrial glacial records. Their stochastic modeling starts with descriptions on different spatial and temporal scales, using mathematical approaches on different levels: microscopic modeling through interacting systems of individual agents or particles, random walks, or random media, and macro-/mesoscopic modeling using stochastic (partial, delay) differential equations, and stochastic dynamics. Typically, the same system can encompass different scales, and it is crucial to focus on the relevant one when specifying a model for the problem of interest and when deriving it through limit theorems. Besides the appropriate choice of scale, for practical use the models need fine tuning by calibration and optimization. Here numerical simulation tools and statistical techniques – such as parameter identification, calibration, regression or dimension reduction – can be used to select best-fitting models.

The Berlin stochastics community, one of the strongest and largest in Germany, is globally visible and active within international and multidisciplinary research networks shaping the key areas of modern stochastics. It encompasses groups at HU, TU, WIAS, and U Potsdam. It has a proven record of successful collaboration in and across its focal areas of research and research training, as demonstrated in the Berlin Probability Colloquium and in research seminars such as “Stochastic Analysis and Stochastic Finance”, Stochastic Models in Physics and Biology", or "Mathematical Statistics”. This radition is also reflected in joint activities in research training, as realized in the former RTG Stochastic Processes and Probabilistic Analysis, and the former IRTG Stochastic Models of Complex Processes and their Applications with ETH and U Zürich. In the past four years, the group underwent a substantial renewal reflecting the strong dynamism of all areas of stochastics: 10 out of 16 faculty members have been newly appointed, including 4 junior professors. The new colleagues have strengthened and re-focused domains of activity, and created new core areas of research dealing with trading in illiquid markets, volatility risk and affine models, rough paths theory, the stochastic dynamics of coupled neural networks, random walks among random conductances, and biostatistics.The resulting momentum in the Berlin stochastics groups has triggered an initiative for new RTG Stochastic analysis and its applications in biology, finance and physics.

Here we outline the main directions of research of the Berlin groups in stochastics. There are strong links to other areas, in particular to Numerical mathematics and scientific computing and to Applied analysis and differential equations.

Stochastic analysis and dynamics (Friz, Imkeller, Scheutzow): Mathematical models in this area arise for instance in the study of high frequency financial or climate data, or in optimal control, and are described by (backward) S(P)DE. We shall apply the tools of BS(P)DE and rough path analysis to new domains, study geometric and invariant structures of stochastic flows, and investigate ergodic theory for SPDEs, like invariant structures for the stochastic Navier-Stokes equation with Lévy noise. The study of metastability patterns of stochastic dynamical systems (arising for instance in climate dynamics and neuroscience) will be combined with model calibration for paleo-climatic real time series.

Stochastic processes in physics and biology (Blath, Deuschel, König, Roelly): Probabilistic models in mathematical genetics explain patterns of genetic variability among populations by the interplay between evolutionary forces such as genetic drift, selection, or recombination. In mathematical population dynamics, we study macroscopic aspects (such as longtime behaviour, coexistence of several species, ‘survival of the fittest’, and the meta-stability of populations with multiple characteristics) by means of spatial stochastic systems with various branching mechanisms. In computational neuroscience, we investigate the stochastic dynamics of coupled neural networks. The parabolic Anderson model is a prominent example of random motion in random media. Research here deals with the interplay between random potentials and random walks among random conductances. Large classical and quantum systems with pair interactions are examples of interacting many-body systems; here we investigate the emergence of crystalline structures in cold dilute systems or the cycle structure in bosonic systems. We aim to extend our study of phase transitions in random interface models to other models from statistical physics.

Statistical Inference (Dickhaus, Reiss, Spokoiny): We develop adaptive smoothing methods with applications to medical imaging and volatility estimation, and the statistics or stochastic processes in risk management, monitoring of high frequency markets, model reduction, and model selection. For continuous time jump processes we build p a general theory, unifying high- and low-frequency approaches, for inference in jump diffusion and semimartingale models, significant in physics, biology, and especially finance. We shall develop procedures for testing multiple hypotheses able to cope with data-analytic challenges in biometrics, (genetic) epidemiology and neuroscience, especially with the “curse of dimensionality”, and complex noise structures.

Stochastic finance (Bank, Becherer, Horst, Keller-Ressel, Kupper): Focusing on optimization, hedging and equilibrium in incomplete markets, we combine techniques of stochastic analysis and the stochastic calculus of variations to approach optimization problems by means of systems of forward–backward stochastic differential equations. We include market friction and illiquidity effects caused for instance by the impact of large traders in new modeling approaches practically relevant for optimal risk management in illiquid markets. To deal with risk caused by effects such as stochastic volatility or jumps in sset prices, we investigate affine stochastic volatility and term structure models. These are significant for volatility derivatives such as variance swaps, and also for their computational tractability.