Address:Institute for Mathematics
Royal Institute of Technology
Lindstedts väg 25
100 44 Stockholm
Sweden
You can find me in the Room 1646 using entrance Osquars backe 2.
Postal Address:
Matematik, KTH
S-100 44 Stockholm
Telephone: +46 - 8 - 790 - 7160
Email: cbayer@kth.se
Research:
My main research interest is numerical methods for stochastic differential equations (SDEs).
In 2004, T. Lyons and N. Victoir proposed a new method for approximating the solution of the Kolmogorov backward equation corresponding to a given SDE. The Kolmogorov backward equation is a parabolic second order PDE which can be approximated by Monte-Carlo simulation using its stochastic representation in terms of the solution to the SDE. Even though Lyon's and Victoir's "Cubature on Wiener"-method uses the stochastic representation, it is a deterministic method with deterministic a-priori error bounds. It retains the main advantage of Monte-Carlo methods as it does not suffer from the "curse of dimensionality". In principle, cubature-methods of any order can be constructed, but in numerical practice there are still severe restrictions and a lot works remains to be done, from a numerical as well as from a more theoretic point of view. In particular, I would like to extend the method to higher order differential operators and I work on a reduction of the cubature paths using rotation invariance of Brownian motion. I think that "Cubature on Wiener space" is especially interesting since it has links to many different subjects such as stochastic analysis, differential geometry, numerical mathematics and algebra.
The theory behind "Cubature on Wiener space" is closely linked to some stochastic process on a free nilpotent Lie group depending on the number of Brownian motions and the order of the method (more precisely, the aforementioned stochastic process is the solution of the martingale problem associated to the sub-Laplacian associated to the sub-Riemannian geometry on the Lie group). Approximations of the heat kernel on free nilpotent Lie group can lead to new numerical methods. I try to use this idea in order to find feasible Milstein-type schemes for SDEs driven by more than one Brownian motion and it might also be possible to apply this idea to the calculation of Greeks in financial mathematics.
The third focus of my work is on numerical methods for reflected SDEs. Reflected SDEs provide stochastic representations for parabolic PDEs as above with Neumann boundary conditions. The usual Euler-Monte-Carlo method works also for reflected SDEs (with some modifications due to the reflection), but the error converges with order 1/2 - ignoring the additional error from the Monte-Carlo simulation. Together with A. Szepessy and R. Tempone, I work on faster methods for reflected SDEs, in particular by using adaptive meshes.
Diploma thesis: Cubature on Wiener space extended to higher order operators (pdf); supervisor: Josef Teichmann
PhD thesis: Selected topics in numerics of stochastic differential equations (pdf); supervisor: Josef Teichmann
Publications and Preprints:
1) Christian Bayer, Josef Teichmann: The proof of Tchakaloff's Theorem (pdf), Proc. Amer. Math. Soc. 134 (2006) 3035-3040.
2) Christian Bayer: Brownian Motion and Itô Calculus (pdf), Lecture notes from a short course given at the WK summer camp 2006.
3) Christian Bayer: The Geometry of Iterated Stratonovich Integrals (pdf), preprint 2006.
4) Christian Bayer, Josef Teichmann: Cubature on Wiener space in infinite dimension (pdf), Proceedings of the Royal Society A, 464(2097), 2008.
Presentations:
1) Discretization of SDEs: Euler Methods and Beyond (pdf). Talk given at the PRisMa 2006 One-Day Workshop on Portfolio Risk Management, Vienna, Austria.
2) Calculation of the Greeks Using Cubature Malliavin Calculus (pdf). Talk given at FSU, Tallahassee, Florida.
3) Cubature for infinite-dimensional SDEs with applications to interest rate modelling (pdf). Talk given Fifth World Congress of the Bachelier Finance Society, London, 2008.
4) Weak adaptive approximation of reflected diffusions (pdf). Talk given at the Dahlquist Fellowship Workshop 2008, Stockholm.
Posters:
1) Prinzip der Versicherung (pdf). Poster created for the "Lange Nacht der Forschung" (in German, A4-version).
2) Rückversicherung und Katastrophenbonds (pdf). Poster created for the "Lange Nacht der Forschung" (in German, A4-version).
3) Cubature for infinite dimensional SPDEs (pdf). Poster presented at the AMaMeF Conference 2007 in Vienna, Austria.
Research visits, workshops and talks: