The Berlin-Potsdam stochastics and mathematical finance community consists of researchers at the Berlin universities FU, HU, TU, as well as U Potsdam, and the Weierstrass Institute (WIAS). It is globally visible and highly active at forefronts of many key areas of modern probability theory, with applications in various fields. The group participates in and shapes several national and international research networks. Brisk exchange occurs in several regular research seminars on subdomains like SPDEs, stochastic analysis for finance, and stochastic models in biology and physics. Frequently organized high-class scientific events like workshops, congresses and schools regularly bring many of the world-leading experts to the Berlin-Potsdam area.

**Research training**

Our vision and mission is to train a new generation of researchers in the vibrant field of stochastic processes that presents formidable theoretical challenges in its mathematical foundations and that breathes and grows through its applications.

The community has a long-term record of successful collaboration in carrying out structured re-search training and organizing PhD education programs. This is demonstrated by the awards and successful realizations of a consecutive series of DFG-funded graduate schools over the last three decades. Currently, we are running the

Berlin-Oxford International Research Training Group (IRTG) 2544 „Stochastic Analysis in Interaction“,

a research training initiative of the Berlin universities (FU, HU, and TU Berlin) and the WIAS, jointly with the University of Oxford.

The IRTG is a certified unit of the Berlin Mathematical School (BMS). Through the BMS we are also part of the Berlin Mathematics Research Center Math+.

Previous (I)RTGs include

RTG “Stochastic Processes and Probabilistic Analysis” (1996-2004)

IRTG “Stochastic Models of Complex Processes” (2006-11) with ETH and U Zürich

RTG “Stochastic Analysis with Applications in Biology, Finance, and Physics” (2012-17)

**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 RTA 6 "Numerical mathematics and scientific computing" and to RTA 7 "Applied analysis and differential equations" and to RTA 8 "Mathematics of data science".

**Stochastic analysis and dynamics**

(Bayer, Becherer, Blath, Deuschel, Friz, König, Paycha, Perkowski, Reiß, Scheutzow, Stannat):

Recent decades saw a revolution in the analytic treatment of stochastic (partial) differential equa-tions (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 con-tinues to be, e.g. through the study of optimal control, backward S(P)DEs, singular SPDEs, er-godic theory for SPDEs, stochastic flows, stochastic dynamical systems, and much more. These topics are closely linked to applications e.g. in climate models, financial markets, neural network analysis, population dynamics and more.

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

(Blath, Deuschel, König, Kurt, Perkowski, Reiß, Roelly, Schwalger, Stannat, Wilke Berenguer):

Several research directions of the Berlin-Potsdam stochastics community are triggered and driv-en by interests in applications in biology, statistical 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, recombination and also dormancy. In population dynamics and ecology, we study mac-roscopic aspects (such as longtime behaviour and coexistence of species) by means of spatial stochastic systems with various branching mechanisms. In neuroscience, we analyze neural ac-tivity in the brain on different scales with the help of stochastic evolution equations and stochas-tic 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 stat-ic 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, inter-acting Poisson point processes are used for modeling spatial wireless telecommunication net-works, 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 estima-tion of model parameters or for testing the validity of model assumptions. This requires to com-bine 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, Bayer, Becherer, Belak, 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 prob-lems 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 deriva-tives such as variance swaps, and also for their computational tractability. We also develop sta-tistical 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 on probability and stochastic processes give an introduction to the most important concepts of modern probability theory. They are fundamental to the theory of stochas-tic processes and their stochastic and statistical analysis and many 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. Good knowledge of the foundations of measure theory and Lebesgue integration.

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

◦ Construction of stochastic processes

◦ Conditional expectations and laws

◦ Martingales: stopping theorem, convergence, inequalities

◦ Markov chains, time series, Poisson process

◦ 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**

Courses are regularly offered on various advanced topics from the spectrum of our interests, including statistics of stochastic processes, mathematical finance, stochastic partial differential equations, rough paths and regularity structures, interacting particle systems, stochastic pro-cesses in evolution, stochastic processes in neuroscience, spatial stochastic point processes, stochastic control, Markov processes, Lévy processes, computational finance, large deviations.