# Christian Bayer

### Teaching

Übung Finanzmathematik 1

Übung Finanzmathematik 2

Vorlesung Computational Finance

Institute of Mathematics
TU Berlin
Strasse des 17. Juni 136
10623 Berlin
Germany

Room MA 711 / 7th floor.
Telephone: +49 - 30 - 314 - 21271
Email: bayer@math.tu-berlin.de

Sprechstunde: Donnerstag, 10:00 - 11:30

### 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 theory behind cubature formulas on Wiener space is closely linked with rough path theory, and I am working with P. Friz on these connections.

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.

I am also interested in symplectic methods for molecular dynamics, based on Ehrenfest and Born-Oppenheimer approximations of the Schroedinger equation.

Together with Klaus Wälde I am working on some mathematical problems in economic modelling. The economic problem consists in understanding an optimal consumption-saving choice in the presence of uncertain labour income where uncertainty is driven by two Poisson processes. From a mathematical perspective, we apply the apparatus of Fokker-Planck equations and other techniques for the analysis of Markov processes to understand the distributional and ergodic properties of this model.

### 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 Itò 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.

5) Christian Bayer, Anders Szepessy, Raul Tempone: Adaptive weak approximation of reflected and stopped diffusions (pdf), submitted 2009.

6) Christian Bayer, Klaus Wälde: General Equilibrium Island-Matching and Saving in Continuous Time: Theory (pdf), in preparation.

7) Christian Bayer, Klaus Wälde: General Equilibrium Island-Matching and Saving in Continuous Time: Proofs (pdf), in preparation.

### 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) Weak adaptive approximation of reflected diffusions (pdf). Talk given at the Dahlquist Fellowship Workshop 2008, Stockholm.

4) Hypo-elliptic simulated annealing  (pdf). Talk given at the SPA 2009  conference in Berlin.

5)  Cubature on Wiener space for Heath-Jarrow-Morton interest rate models (pdf). Talk given at the RIMS workshop on Computational Finance  2009 in Kyoto.

### 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:

• 09/23/2004-09/25/2004: Participation at the Workshop on ALM, Vienna, Austria.

• 04/12/2005: Talk on "An Elementary Proof of Tchakaloff's Theorem" in the FAM-seminar, Vienna, Austria.

• 04/25/2005-06/30/2006: Research visit to Prof. Anders Szepessy at KTH, Stockholm, Sweden.

• 06/27/2005: Talk on "Cubature on Wiener space with an example in finance" in the seminar of NADA, KTH, Stockholm, Sweden.

• 07/08/2005-07/15/2005: Participation at Dimitsana Summer School on Stochastic Differential Geometry and Applications in Finance, Thermon Aitoloakarnanias, Greece.

• 09/19/2005-09/22/2005: Participation at the 16th international Congress of the Austrian Mathematical Society (Ã?MG), Klagenfurt, Austria.

• 11/09/2005-11/12/2005: Participation at AMaMeF Workshop on Stochastic Analysis and Computational Finance, London, UK.

• 01/31/2006-02/03/2006: Participation at the workshop "Numerical Methods in Finance", within the AMaMeF program, Paris, France.

• 05/30/2006-06/16/2006: Research visit to Prof. Anders Szepessy at KTH, Stockholm, Sweden.

• 07/02/2006-09/02/2006: Participation at the WK summer camp at Weissensee, Austria.

• 07/03/2006-07/07/2006: Short course on "Brownian motion and Ito calculus" at the WK summer camp, Weissensee, Austria.

• 09/26/2006: Talk on "Discretization of SDEs: Euler Methods and Beyond" at the PRisMa 2006 One-Day Workshop on Portfolio Risk Management, Vienna, Austria.

• 10/10/2006-11/03/2006: Research visit to Raul  Tempone at FSU, Tallahassee, Florida, US.

• 10/25/2006: Talk on "Calculation of Greeks using Malliavin calculus I" in the seminar of the School of  of Computational Science of the FSU, Tallahassee,Florida.

• 10/26/2006: Talk on "Calculation of Greeks using Malliavin calculus II" in the seminar of the Department of Mathematics, FSU, Tallahassee, Florida.

• 08/09/2007-08/12/2007: Participation at Satellite Summerschool on Levy Processes: Theory and Applications in Sandbjerg Manor, Denmark.

• 08/13/2007-08/17/2007: Poster presentation at 5th International Conference on Levy Processes: Theory and Applications, Copenhagen, Denmark.

• 09/17/2007-09/22/2007: Poster presentation at Workshop and Mid-Term Conference on Advanced Mathematical Methods for Finance, Vienna, Austria.

• 03/04/2008: Talk on "Cubature for infinite-dimensional SDEs" at the 8th German Open Conference on Probability and Statistics, Aachen, Germany.

• 05/27/2008: Talk in the seminar of the group for Probability and Statistics of the Institute for Applied Mathematics, University of Bonn, Bonn, Germany.

• 07/01/2008: Talk in the seminar of the subdivision for statistic of the Institute for Applied Mathematics of the University Heidelberg, Heidelberg, Germany.

• 07/18/2008: Invited talk on "Cubature for infinite-dimensional SDEs" at the Fifth World Congress of the Bachelier Finance Society, London, United Kingdom.

• 10/09/2008: Talk on "Cubature on Wiener space for Heath-Jarrow-Morton interest rate models" in the seminar of the Department of Mathematics, FSU, Tallahassee, Florida.

• 10/20/2008: Talk on "Adaptive weak approximation of reflected diffusions" in the Dahlquist Fellowship Workshop, Stockholm, Sweden.

• 02/11/2009: Talk on "Cubature on Wiener space for Heath-Jarrow-Morton interest rate models" at the PREMIA meeting in Paris, France.

• 06/29/2009: Talk on "Cubature on Wiener space for Heath-Jarrow-Morton interest rate models" at the ENUMATH 2009 conference in Uppsala, Sweden.

• 07/30/2009: Invited talk on "Hypo-elliptic simulated annealing" at the SPA 2009 conference in Berlin, Germany.

• 08/10/2009: Talk on "Cubature on Wiener space for Heath-Jarrow-Morton interest rate models" at the RIMS workshop on "Computational Finance" in Kyoto, Japan.