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Camilli, Fabio  Seminar room C 714  Mon, 12. Nov 07, 14:00 
Homogenization of HamiltonJacobi equations: Numerical Methods  
We study approximation strategies for the limit problem arising in the homogenization of HamiltonJacobi equations. They involve first an approximation of the effective Hamiltonian then a discretization of the HamiltonJacobi equation with the approximate effective Hamiltonian. We give a global error estimate which takes into account all the parameters involved in the approximation.  

Tysk, Johan  Seminar room C 714  Mon, 12. Nov 07, 15:30 
FeynmanKac formulas for BlackScholes type operators  
There are many references showing that a classical solution to the BlackScholes equation is a stochastic solution. However, it is the converse of this theorem that is most relevant in applications, and the converse is also more mathematically interesting. In this talk we establish such a converse. We find a FeynmanKactype theorem showing that the stochastic representation yields a classical solution to the corresponding BlackScholes equation with appropriate boundary conditions under very general conditions on the coefficients. We also study the pricing equation in the presence of bubbles, ie when the price process is a strict local martingale. In this case there is an infinite dimensional space of classical solutions. These results are obtained jointly with Svante Janson and Erik Ekström, respectively.  

Shirikyan, Armen  Seminar room C 714  Tue, 13. Nov 07, 9:30 
Degenerate elliptic equations and stationary measures for 3D stochastic NavierStokes system  
Let us consider 3D NavierStokes (NS) equations perturbed by a degenerate random force. A solution $u(t,x)$ of this problem is a random process in an appropriate functional space. We say that the solution $u$ is stationary if the law of $u(t,cdot)$ does not depend on time. A stationary measure for the NS equations is defined as the law of a stationary solution. The aim of my talk is to present some qualitative properties of stationary measures. Roughly speaking, we show that if the random perturbation is sufficiently nondegenerate, then the support of any stationary measure coincides with the entire phase space, and its finitedimensional projections are minorised by the Lebesgue measure multiplied by a smooth positive density.  

Bardi, Martino  Seminar room C 714  Tue, 13. Nov 07, 11:00 
Multiscale problems for BellmanIsaacs parabolic PDEs  
We survey a general approach to singular perturbations and homogenization problems for HamiltonJacobiBellmanIsaacs 1st and 2nd order equations arising in the reduction of dimension of multiscale control systems. They are formulated for optimal stochastic control problems or for zerosum differential games, via the associated dynamic programming PDEs and their viscosity solutions. In particular, we present results for problems with an arbitrary number of scales and with oscillating terms in the PDE as well as in the initial data. Most of the results are obtained in collaboration with O. Alvarez and C. Marchi.  

Teichmann, Josef  Seminar room C 714  Tue, 13. Nov 07, 14:00 
Natural OUprocesses on Lie groups with applications to simulated annealing  
We show that a natural class of hypoelliptic processes on Lie groups admits an invariant measure and a spectral gap with respect to it. We apply this class of processes to construct simulated annealing algorithms which converge in distribution to minima of nonconvex functionals. The algorithms are nonelliptic and need therefore less independent Brownian motions than space dimensions. The universal constants depend on the geometry of certain nilpotent Lie groups. We apply the DriverMelcher inequalities on Lie groups to show the main estimates.  

Pardoux, Etienne  Seminar room C 714  Tue, 13. Nov 07, 15:30 
Periodic Homogenization : on the homogenized diffusion matrix  
We know how to prove an homogenization result, by a probabilistic method, for the solution $u^\eps$ of an elliptic or parabolic second order PDE with periodic coefficients, even when we allow the matrix of second order coefficients to degenerate, for example to vanish on an open set. In this talk, we will concentrate on the caracterization of the range of the homogenized diffusion matrix (in particular we shall say when this matrix is non degenerate). The results are joint with Martin Hairer (Warwick).  

Djehiche, Boualem  Seminar room C 714  Wed, 14. Nov 07, 9:30 
Systems of variational inequalities with interconnected obstacles A probabilistic approach.  
I will review some recent results on existence of viscosity solutions to systems of variational inequalities with interconnected obstacles, driven by a second order linear operator. We give an equivalent formulation as an optimal multiswitching problem, whose solution is given by solving a system of reflected backward SDEs with oblique reflection. This is joint work with S. Hamadéne.  

Gomes, Diogo  Seminar room C 714  Wed, 14. Nov 07, 11:00 
Generalized AubryMather problem and Stochastic Optimal Control  
In this talk we describe the generalized Mather problem and its connections with stochastic optimal control. Namely, we will establish representation formulas for viscosity solutions and show how these formulas imply uniqueness of solutions.  

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