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Vorlesung Hoehere Wahrscheinlichkeitstheorie

Department of Mathematics

University of Vienna

Nordbergstrasse 15

1090 Wien

Austria

Room C 4.12 / 4^{th}floor.

Telephone: +43 1 4277 50686

Email: christian.bayer@univie.ac.at

Sprechstunde: Freitag, 10:15 - 11:15

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 Waelde 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.

Cubature on Wiener space extended to higher order operators (

Selected topics in numerics of stochastic differential equations (

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 Ito 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), Monte Carlo Methods and Applications 16 (2010), 1--67.

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

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

8) Christian Bayer, Peter Friz, Ronnie Loeffen:Semi-closed form cubature and applications to financial diffusion models(pdf), submitted.

9) Christian Bayer, Peter Friz:Cubature on Wiener space: Pathwise convergence(pdf), submitted.

10) Christian Bayer, Klaus Waelde:Existence, Uniqueness and Stability of Invariant Distributions in Continuous-Time Stochastic Models or: Matching and Saving in Continuous Time: Stability(pdf), preprint.

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.

6) Some applications of cubature on Wiener space (pdf). Talk given at the WIAS 2011.

1) Prinzip der Versicherung (pdf). Poster created for the "Lange Nacht der Forschung" (in German, A4-version).

2) Rueckversicherung 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.