Probability, Statistics and Stochastic Analysis
Stochastics is the mathematical theory of randomness. The exact mathematical description of random processes in nature, society, economy and technology is the necessary prerequisite for their deeper understanding and their quantitative description forms the basis of many technical applications. Mathematical finance focuses on the mathematical description and analysis of financial markets.
The mathematical objects considered are stochastic processes (in space and time), that for example can be described in terms of stochastic ordinary or partial differential equations. Stochastic analysis forms the central mathematical theory for their qualitative and quantitative analysis, as well as their numerical approximation.
An important problem area for stochastics is the analytical description of statistics of random processes using limit theorems (laws of large numbers, central limit theorems, mean-field limits, large deviations, ...) for underlying microscopic random systems, such as those arising in statistical physics in the form of interacting particle systems; in economics in the form of interacting market participants; or in the social sciences in the context of social networks. In comparison with classical continuum mechanics, stochastics also provides methods for the rigorous mathematical description of asymptotic fluctuations and approximation errors. Conversely, following the example of evolution in nature, stochasticity, as a mathematical principle, can help develop new mathematical algorithms or improvie existing ones, for example in search problems or optimization.
Rough analysis, inspired by rough path theory, provides a nonlinear extension of distribution theory to analyze singular stochastic dynamics and nonlinear signal effects, with significant impact on stochastic partial differential equations. It has evolved to reveal profound geometric and algebraic structures.
Berlin research groups
| Peter Bank | Stochastic Analysis and Control in Mathematical FinanceThe inherent randomness of financial markets makes Probability Theory an indispensable tool for their description and for furthering our understanding of this crucial aspect of our society. Most notably, Stochastic Analysis and Stochastic Control Theory are instrumental for describing optimal ways to, e.g., mitigate risk or allocate financial resources in an optimal manner. Peter Bank’s group develops novel models and methods in these areas of mathematics and shows how to bring them to bear on financial-economic challenges. |
TU Berlin |
Stochastic AlgorithmsMotivated from modern models in mathematical finance (rough volatility) and from machine learning applications, one research focus is the development and analysis of computational methods for option pricing, optimal stopping, and stochastic optimal control, when the underlying state process is not a Markov process. The (rough path) signature is a canonical tool for approximation of functions on a path, and, hence, ideally suited for numerical approximations of functionals of a process with memory. Additional core research questions concern stochastic optimal control problems under uncertainty and high-dimensional numerical integration. |
WIAS | |
Stochastic Analysis and Stochastic FinanceOptimal stochastic control, martingale methods, singular control, mean-field games, forward/backward stochastic differential equations, rough stochastic differential equations, discontinuous stochastic dynamics, applications thereof (e.g. hedging and optimal trade execution, multi-player games with relative performance concerns, mean field games of information access). |
HU Berlin |
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| Ana Djurdjevac | Numerical Analysis and StochasticsWe are interested in the analysis and numerical analysis of stochastic partial differential equations (PDEs), uncertainty quantification and surface PDEs on time-dependent domains. We consider various mathematical problems that arise from the connection of mathematics with other sciences such as biology, physics and life sciences, where (S)PDEs are used as models. Particularly, we are studying well-posedness, stability and regularity results, which can serve as a basis for numerical analysis and computations. |
FU Berlin |
Rough and Stochastic AnalysisPeter Friz is interested in all aspects of rough and stochastic analysis and this includes topics such as rough paths and regularity structures, Malliavin calculus, fast-slow systems and homogenization, time series and signatures, classes of SPDEs and applications to quantitative finance, notably in rough volatility. There are collaborations with WIAS (Bayer, Liero), with Becherer and Horst (HU Berlin) and Djurdjevac and Perkowski (FU Berlin). |
TU Berlin and WIAS | |
Stochastic Control and Game Theory We do interdisciplinary research at the interface of mathematical finance, probability theory and game theory. Our focus is on stochastic optimal control theory, mean-field and stochastic games, and scaling limits for stochastic processes, especially Hawkes processes. Applications of our research in finance and economics include problems of optimal trade execution in illiquid financial markets, problems of optimal exploitation of exhaustible resources, equilibrium theory, and rough volatility models. |
HU Berlin |
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Discrete Stochastic ModelsKönig is interested in various critical phenomena occurring in discrete spatial random models of many kinds, like random graphs embedded in space, random point processes with interactions and the resulting Gibbs measures, particle processes in space with chemical interactions like coagulation. The reserach group studies the system in the large-term limit with methods from asymptotic probability (e.g., large deviations) and analysis (e.g., variational analysis) and combinatorics. Applications stem from statistical mechanics, chemical engineering and telecommunication. |
TU Berlin and WIAS | |
| Dörte Kreher | Modelling of Financial MarketsThe market microstructure of financial markets can be described via complex stochastic dynamics. It is an active direction of research to derive and analyze approximating SDE/SPDE models via scaling limits. Furthermore, models for asset price bubbles in discrete and continuous time are studied. |
HU Berlin |
Stochastic analysis and singular dynamicsOur research activities lie in the area of stochastic analysis with main interest in the development of new mathematical methods for stochastic dynamics. This is motivated by applications in mathematical physics, biology or finance. A guiding theme is the notion of scales, for example new mathematical methods are required to control small scale singularities in nonlinear SPDEs with irregular noise and to remove possible resonances by renormalization, and it is an intriguing problem to derive effective dynamics for the macro- and mesoscopic behavior of complex systems. Conversely, irregular noise can regularize otherwise ill-posed deterministic systems, which is a fascinating area of research bridging stochastic and deterministic analysis. |
FU Berlin |
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Mathematical StatisticsIn mathematical statistics we ask about the construction and the properties of statistical methods like estimators, tests or classifiers. Based on a rigorous modelisation the aim is to find optimal procedures among all feasible ones (which may be as large as all measurable functions of the data). The statistical problems considered vary from modern high-dimensional statistics and machine learning where the data sets and the unknown parameters are huge over nonparametric statistics and inverse problems where the unknowns are functions to statistics of stochastic processes where realisations from stochastic dynamics, e.g. described by stochastic (partial) differential equations, generate the data to work with. Modern statistics does not only require advanced knowledge of probability theory, but has also strong relationships with functional analysis, optimisation and numerical analysis as well as with modern developments in data science and applied fields. A mathematical understanding of statistical problems and methods is key for correct usages and poses at the same time challenging fundamental questions. |
HU Berlin |
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Stochastic Processes in NeuroscienceIn our research, we want to understand how the dynamics of neural networks in the brain emerges from the complex interplay of thousands of nerve cells. To this end, we develop and analyze stochastic models of neural activity across different scales, starting from point process models of single neurons at the microscopic scale and systematically coarse-graining them to stochastic partial differential or integral equations that describe the dynamics and fluctuations of neural populations at the mesoscopic and macroscopic scale. For this multiscale approach, our group uses mathematical tools from statistical physics, stochastic processes and nonlinear dynamics, such as mean-field theory, central limit theorems and dimensionality reduction techniques. Specific research topics include spectral decompositions of stochastic Fokker-Planck equations, statistical inference methods for mesoscopic neural data, stochastic neural field equations and non-renewal sequences of first-passage times. Our mathematical framework will enable us to better understand neural variability in cortical circuits in terms of their biophysical mechanisms and their functional role for information processing. |
TU Berlin | |
Stochastic Algorithms and Nonparametric Statisticsno text available yet |
HU Berlin and WIAS |
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Stochastic Partial Differential Equations: Theory and Applications Stochastic partial differential equations (spde) are the mathematical framework to model the continuous time evolution of spatially extended random dynamical systems. Their scope of applications ranges from more classical applications in the engineering and natural sciences (dynamical state and parameter estimation, fluctuations in continuum limit approximations of physical systems, like e.g. stochastic reaction-diffusion systems or stochastic Navier-Stokes equations) to the modelling and analysis of neural activity in the brain on mezoscopic and macroscopic scales. Particular aspects in our research agenda are: stochastic filtering, stochastic optimal control and mean-field theory for and with spde. |
TU Berlin | |
Mathematical and statistical foundations of data scienceIn medical imaging or weather prediction, how does one optimally combine real-world data with knowledge from computer-based PDE models? How does one generate samples from high-dimensional and complicated probability distributions? Can machine learning models approximate complex non-linear phenomena (such as PDE dynamics) without suffering a "curse of dimensionality"? How can mathematical and statistical models help us think about, and improve, democratic decision making mechanisms? Mathematically, the above questions can be formalized as statistical and algorithmic problems which are typically high- or infinite-dimensional, and subject to complex structural information. This is an exciting and rapidly growing area which uses tools of statistics, probability, analysis, PDE and computation, with many open theoretical and applied problems. Topics include Bayesian inference for inverse problems and PDE models, statistical procedures for diffusion data, mixing times for Markov Chain Monte Carlo (MCMC) methods, statistical theory for generative models such as NeuralODEs, and algorithms for the random selection of representative citizens’ assemblies. |
HU Berlin |
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Interdisciplinary MathematicsWe consider problems involving stochastic processes inspired by other disciplines. Currently the focus is mathematical population genetics where we study the phenomena related to (the modelling of) evolution. |
HU Berlin |
Links to other areas
There are natural, strong collaborative ties to the following groups:
Partial Differential Equations - Matthias Liero, Marita Thomas, Barbara Zwicknagl
Numerical Methods - Claudia Schillings
Algebra and Geometry in Data Science - Carlos Améndola
Core Courses
The BMS Core Courses in Area 3 and details about their content can be found under "Course Program":
Area 3 - Core Courses
Advanced Courses
Topics for typical advanced courses include:
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.
The current advanced courses are listed under "Course Program":
Area 3 - Advanced Courses
