Focus platform: Quantitative analysis of stochastic and rough systems
Stochastics plays an increasingly central role in numerous scientific as well as engineering problems, both as a powerful computational tool - multi-level Monte Carlo methods, stochastic gradient descent, expectation maximization (EM) algorithm are just few of many relevant examples - and as a model of uncertainties inherent in complex phenomena.
The aim of this group is to conduct research at the intersection and forefront of the afore-mentioned fields.
For instance, analysis of financial and energy markets has recently seen the need to go beyond the paradigm of Markovian stochastic processes - itself a consequence of (over)simplified models based on aggregation of independent random shocks over time. In many application areas that noise exhibits negative auto-correlations, leading to inherent roughness in corresponding stochastic models, poorly handled with standard theory of stochastic analysis. With concrete motivation from rough volatility, a major development in the field, investigated within a joint DFG project (WIAS-PI: Christian Bayer) and also the strand (ii) of a ERC-CoG project (WIAS-PI: Peter K. Friz), there is a clear outlook to develop a full range of methodology related to the analysis and numerical treatment of non-Markovian, non-martingale based stochastic systems. This relates naturally to Terry Lyons' rough path theory, a purely analytical view on stochastic theory based on higher-order numerical schemes. After years of foundational work, the last few years have seen multiple breakthroughs, especially in the field of stochastic partial differential equations, itself tied to concrete applications such as stability of non-linear filtering problems or surface growth modelling. Numerical and algorithmic aspects of resulting classes of rough partial differential equations is one of the goals of DFG research unit FOR 2402 / project P3 (PI: Christian Bayer, John Schoenmakers) situated at WIAS.
Of course, rough path and related methods are only one toolsets useful for a proper analysis, control and uncertainty quantification (UQ) of large, complex systems, may this related to energy or financial markets. Future modeling is very likely to be data-driven, hence new methods of data-science (machine learning, deep learning, ...) are crucial. (In his 2014 ICM lecture, Terry Lyons points out several relations to rough paths and signatures; already used for an award-winning Chinese character recognition algorithm.) In addition to afore-mentioned classical inference methods, much emphasis will be given to develop these new ideas, in and beyond the mentioned applications fields.
We are grateful for the funding provided by:
- DFG individual grant "Rough stochastic volatility and related topics" (BA 5484/1-1), started in 2015.
- DFG Research Unit FOR 2402 "Rough Paths, Stochastic Partial Differential Equations and Related Topics" - project P3 together with Raphael Kruse (TU Berlin) and John Schoenmakers (WIAS), started in 2016.
- ERC Consolidator Grant Geometric aspects in pathwise stochastic analysis and related topics, started in 2016.
- Peter Friz, Martin Hairer: A Course on Rough Paths, Springer, 2014.
- Christian Bayer, Peter Friz, Jim Gatheral: Pricing under rough volatility (pdf), Quantitative Finance, 16(6), 887-904, 2016.
- Christian Bayer, Peter Friz, Sebastian Riedel, John Schoenmakers: From rough-path estimates to multi-level Monte Carlo (pdf), SIAM J. Numer. Anal., 54(3), 1449-1483, 2016.
- Felix Anker, Christian Bayer, Martin Eigel, Marcel Ladkau, Johannes Neumann, John Schoenmakers: SDE based regression for linear random PDEs (pdf), SIAM Journal on Scientific Computing, to appear.