Research Group "Stochastic Algorithms and Nonparametric Statistics"
Seminar "Modern Methods in Applied Stochastics and Nonparametric Statistics" Summer Semester 2021
|13.04.21||Yangwen Sun (Humboldt-Universität zu Berlin)|
|Please note: At 4 pm due to time shift!||Graph-spanning ratio test with application to change-point detection problem |
|20.04.21||Caroline Geiersbach (WIAS Berlin)|
| Stochastic approximation with applications to PDE-constrained optimization under uncertainty
|27.04.21||Peter Friz (WIAS Berlin, TU Berlin)|
|New perspectives on rough paths, signatures and signature cumulants |
We revisit rough paths and signatures from a geometric and "smooth model" perspective. This provides a lean framework to understand and formulate key concepts of the theory, including recent insights on higher-order translation a.k.a. renormalization of rough paths. This first part is joint work with C. Bellingeri (TU Berlin), and S. Paycha (U Potsdam). In a second part, we take a semimartingale perspective and more specifically analyze the structure of expected signatures when written in exponential form. Following Bonnier-Oberhauser (2020), we call the resulting objects signature cumulants. These can be described - and recursively computed - in a way that can be seen as unification of previously unrelated pieces of mathematics, including Magnus (1954), Lyons-Ni (2015), Gatheral and coworkers (2017 onwards) and Lacoin-Rhodes-Vargas (2019). This is joint work with P. Hager and N. Tapia.
|11.05.21||Paul Hager (TU Berlin)|
|Optimal stopping with signatures |
We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process. We consider classic and randomized stopping times represented by linear and non-linear functionals of the rough path signature associated to the underlying process, and prove that maximizing over these classes of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature. By applying a deep neural network approach to approximate the non-linear signature functionals, we can efficiently solve the optimal stopping problem numerically. The only assumption on the underlying process is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion, which fail to be either semi-martingales or Markov processes, and can be used, in particular, for American-type option pricing in fractional models, e.g. on financial or electricity markets. This is a joint work with Christian Bayer, Sebastian Riedel and John Schoenmakers.
last reviewed: May 4, 2021 by Christine Schneider