Christian Bayer - Personal Homepage

I am working within the research group Stochastic Algorithms and Nonparametric Statistics at the Weierstraß Institute for Applied Analysis and Stochastics.

I am PI of the focus platform Quantitative analysis of stochastic and rough systems within the Weierstrass Institute.

Research Interests

My main research interests are financial mathematics and stochastic numerics.

In finance, one of my major current research project (with Jim Gatheral, Peter Friz and Benjamin Stemper) is about modeling stock indices like the S & P 500 index (SPX) consistently with respect to the implied volatility surface, and the volatility index (VIX). We find that a surprisingly simple model using a stochastic volatility component involving a fractional Brownian motion allows for great fit of model prices with market option prices using only three parameters. The model is non-Markovian, which leads to significant numerical problems. Together with Peter Friz, I was PI of a DFG project on "Rough stochastic volatility and related topics" (BA 5484/1-1).

A second research interest is the numerical approximation of partial differential equations with random coefficients using stochastic representations based on stochastic (ordinary) differential equations and regression in the spacial variable, in collaboration with Martin Eigel and John Schoenmakers. I also want to study these techniques for partial differential equations driven by random or deterministic rough paths. The advantage of this method is that it enables us to use well known techniques on numerical simulation of diffusion processes and on regression to numerically approximate a much more complicated object. This research is supported by the Research Unit FOR 2402 funded by the DFG.

I am also interested in Monte Carlo algorithms for various more complicated problems. In the past, I have worked on reflected diffusions and on establishing heuristic, efficient and reliable criteria for the choice of the number of samples in general Monte Carlo procedures.

An important research problem in computational finance is numerical approximation of stochastic optimal control problems, in particular optimal stopping. I am collaborating with Denis Belomestny, Raul Tempone, John Schoenmakers, and others on these topics. This strand of research is supported by the DFG via the Berlin Mathematics Research Center MATH+, project AA4-2.

The theory of rough paths has many applications in the field of machine learning. I am, in particular, interested in applications to stochastic optimal control. However, methods from rough path analysis can also be used for theoretical analysis of the properties of deep neural networks. I collaborate with Peter Friz on this subject, in a project supported by the DFG via the Berlin Mathematics Research Center MATH+, project EF1-5/EF1-13.

I am part of the DFG International Research Training Group IRTG 2544 Stochastic Analysis in Interaction, supervising the PhD student Simon Breneis jointly together with Terry Lyons as co-supervisor.

Publications and preprints

Selected presentations

Lecture notes and other short manuscrips

My theses