The Berlin-Potsdam stochastics community consists of researchers at HU, TU, U Potsdam, and WIAS. It is globally visible and highly active at forefronts of many key areas of modern Stochastics with applications in various fields. It participates in and shapes several national and international research networks of various sizes. Brisk exchange occurs in several regular research seminars on subdomains like Finance and SPDEs, and on applications in Biology and Physics. Frequent organisations of high-class scientific events like workshops, congresses and schools of various sizes regularly invite many of the world experts to the Berlin-Potsdam area.

**Research training**

The community has a long-term record of successful collaboration in carrying out structured research training and organizing PhD education programmes. This is demonstrated in the awards and successful realizations of various third-party funded graduate schools ((international) research training groups, (I)RTGs) since the 1990s, like the RTG “Stochastic Processes and Probabilistic Analysis” (1996-2004), the IRTG “Stochastic Models of Complex Processes” (2006-11) with Zürich, the RTG “Stochastic Analysis with Applications in Biology, Finance, and Physics” (2012-17), and the IRTG “High-Dimensional Nonstationary Time Series” (since 2013).

In recent years, the group underwent a substantial renewal reflecting the strong dynamism of all areas of stochastics: ten out of sixteen faculty members have been newly appointed, including four junior professors. The new colleagues have strengthened and refocused domains of activity and have created new core areas of research dealing with trading in illiquid markets, volatility risk, and affine models, rough paths theory, stochastic dynamics of coupled neural networks, random walks among random conductances, and biostatistics.

**Research directions**

Here we outline the main directions of research of the Berlin-Potsdam groups in probability and stochastic processes. There are strong links to other areas, in particular to 6. Numerical mathematics and scientific computing and to 7. Applied analysis and differential equations.

**Stochastic analysis and dynamics**

(Becherer, Deuschel, Friz, Imkeller, König, Paycha, Perkowski, Scheutzow, Stannat):

Recent decades saw a revolution in the analytic treatment of stochastic (partial) differential equations (S(P)DEs) with the help of rough-path theories, and the Berlin-Potsdam community works on the forefront of this development. This is exemplified in the award of two of the prestigious ERC grants and a DFG Research Unit on this topic. Mathematical investigation of SPDEs is a core expertise of the community for decades and continues to be, e.g. in the study of optimal control, climate models, financial markets, neural network analysis, and more. The community studies backward SDEs, stochastic equations arising in mathematical fluid dynamics, singular SPDEs, ergodic theory for SPDEs, stochastic flows, stochastic dynamical systems, and much more.

**Stochastic processes in the natural sciences and engineering sciences**

(Blath, Deuschel, König, Kurt, Reiß, Roelly, Stannat):

Several research directions of the Berlin-Potsdam stochastics community are triggered and driven by interests in applications in Biology, Physics, Neuroscience, Telecommunication or Chemistry. For example, 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 population dynamics, we study macroscopic aspects (such as longtime behaviour, coexistence of several species, "survival of the fittest" and the metastability of populations with multiple characteristics) by means of spatial stochastic systems with various branching mechanisms. In Neuroscience, we analyze neural activity in the brain on different scales with the help of stochastic evolution equations and stochastic neural networks. In many models of random media there arise questions about their global behaviour under the influence of microscopic random mechanisms, like the effective long-time movement of random particles in the medium or about emergent geometric properties or the dichotomy between localised and homogenised behaviour of spectra of random operators or of solutions to random PDEs in that medium. In Statistical Physics and Chemical Physics, large static interacting particle systems, classical as well as quantum, like many-body systems with pair potentials give rise to questions about phase transitions (in terms of infinite-space Gibbs measures) or further effects like condensation and crystallization. In Stochastic Geometry, interacting Poisson point processes are used for modeling spatial wireless telecommunication networks, and percolation problems arise in the study of their connectivity and capacity problems. A key step in stochastic modeling is to construct and analyze statistical methods for the estimation of model parameters or for testing the validity of model assumptions. This requires to combine advanced techniques from stochastic analysis and mathematical statistics, for instance in our current research focus on statistics for SPDEs with applications in physics and neuroscience.

**Stochastic finance**

(Bank, Becherer, Friz, Horst, Kreher, Reiß):

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 liquidity 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 asset 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. We also develop statistical methods for high-frequency transaction prices and limit order book data in order to better understand the underlying market dynamics and to calibrate financial models.

Basic Courses

The two Basic Courses in probability and stochastic processes give an introduction to the most important concepts of modern probability theory. They are fundamental to the theory of stochastic processes and their stochastic and statistical analysis and all application fields. The first course focuses on limiting theories and stochastic processes in discrete time like Markov chains and martingales, but also introduces the one-dimensional Brownian motion. The second gives a solid introduction to continuous-time stochastic processes and the basics of stochastic calculus.

Prerequisites: Basic notions and models in probability theory, the Law of Large Numbers, Central Limit Theorem and elements of estimation and test theory.

Outline of the contents: **Stochastic processes I: discrete time**

◦ Construction of stochastic processes

◦ Conditional expectations and laws

◦ Martingales: convergence, inequalities, optimal stopping

◦ Markov chains and time series

◦ Weak convergence, Brownian motion, and invariance principles**Stochastic processes II: continuous time
**

◦ Brownian motion and continuous-time martingales

◦ Semimartingales

◦ The Itô stochastic integral

◦ Stochastic calculus

◦ Stochastic differential equations

**Spectrum of Advanced Courses**

Regularly offered advanced courses include Statistics of Stochastic Processes, Mathematical Finance, Stochastic Partial Differential Equations, Rough Paths and Regularity Structures, Interacting Particle Systems, Stochastic Processes in Evolution, Stochastic Processes in Neuroscience, Stochastic Control, Markov Processes, Lévy Processes, Computational Finance, Large Deviations.