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Vorlesung Hoehere Wahrscheinlichkeitstheorie
University of Vienna
C 4.12 / 4th floor.
Telephone: +43 1 4277
Freitag, 10:15 - 11:15
research interest is numerical methods for stochastic differential
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.
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
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
on Wiener space extended to higher order operators (pdf);
topics in numerics of stochastic differential equations (pdf);
Publications and Preprints:
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
3) Christian Bayer: The Geometry of Iterated
Stratonovich Integrals (pdf),
4) Christian Bayer, Josef
Teichmann: Cubature on Wiener space in infinite
Proceedings of the Royal Society A, 464(2097), 2008.
Christian Bayer, Anders
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),
7) Christian Bayer, Klaus
Waelde: General Equilibrium Island-Matching and Saving
in Continuous Time: Proofs (pdf),
Friz, Ronnie Loeffen: Semi-closed form cubature and applications to financial
Friz: Cubature on Wiener space: Pathwise
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),
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.
Hypo-elliptic simulated annealing (pdf).
Talk given at the SPA 2009 conference in Berlin.
Cubature on Wiener space for Heath-Jarrow-Morton interest rate models
(pdf). Talk given at the RIMS
workshop on Computational Finance 2009 in Kyoto.
Some applications of cubature on Wiener space
(pdf). Talk given at the WIAS 2011.
Prinzip der Versicherung (pdf).
Poster created for the "Lange Nacht der Forschung" (in
2) Rueckversicherung und
Katastrophenbonds (pdf). Poster
created for the "Lange Nacht der Forschung" (in German,
3) Cubature for infinite dimensional SPDEs
(pdf). Poster presented at the
AMaMeF Conference 2007 in Vienna, Austria.