Upper tails of self-intersection local times of random walks: survey of proof techniques

The asymptotics of the probability that the self-intersection local time of a random walk on $\Z^d$ exceeds its expectation by a large amount is a fascinating subject because of its relation to some models from Statistical Mechanics, to large-deviation theory and variational analysis and because of the variety of the effects that can be observed. However, the proof of the upper bound is notoriously difficult and requires various sophisticated techniques. We survey some heuristics and some recently elaborated techniques and results. This is an extended summary of a talk held on the CIRM-conference on {\it Excess self-intersection local times, and related topics} in Luminy, 6-10 Dec., 2010.Comment: 11 page

How to make Dupire's local volatility work with jumps

There are several (mathematical) reasons why Dupire's formula fails in the non-diffusion setting. And yet, in practice, ad-hoc preconditioning of the option data works reasonably well. In this note we attempt to explain why. In particular, we propose a regularization procedure of the option data so that Dupire's local vol diffusion process recreates the correct option prices, even in manifest presence of jumps

On adaptive estimation in partial linear models

The problem of estimation of the finite dimensional parameter in a partial linear model is considered. We derive upper and lower bounds for the second minimax order risk and show that the second order minimax estimator is a penalized maximum likelihood estimator. It is well known that the performance of the estimator is depending on the choice of a smoothing parameter. We propose a practically feasible adaptive procedure for the penalization choice. (orig.)SIGLEAvailable from TIB Hannover: RR 5549(371)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Estimation of a function with discontinuities via local polynomial fit with an adaptive window choice

New method of adaptive estimation of a regression function is proposed. The resulting estimator achieves near optimal rate of estimation in the classical sense of mean integrated squared error. At the same time, the estimator is shown to be very sensitive to discontinuities or change-points of the underlying function f or its derivatives. For instance, in the case of a jump of a regression function, beyond the interval of length (in order) n"-"1 log n around change-points the quality of estimation is essentially the same as if the location of this jump were known. The method is fully adaptive and no assumptions are imposed on the design, number and size of jumps. The results are formulated in a non-asymptotic way and can be therefore applied for an arbitrary sample size. (orig.)Available from TIB Hannover: RR 5549(291)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Optimal pointwise adaptive methods in nonparametric estimation

The problem of optimal adaptive estimation of a function at a given point from noisy data is considered. Two procedures are proved to be asymptotically optimal for different settings. First we study the problem of bandwidth selection for nonparametric pointwise kernel estimation with a given kernel. We propose a bandwidth selection procedure and prove its optimality in the asymptotic sense. Moreover, this optimality is stated not only among kernel estimators with a variable kernel. The resulting estimators is optimal among all feasible estimators. The important feature of this procedure is that no prior information is used about smoothness properties of the estimated function i.e. the procedure is completely adaptive and 'works' for the class of all functions. With it the attainable accuray of estimation depends on the function itself and it is expressed in terms of 'ideal' bandwidth corresponding to this function. The second procedure can be considered as a specification of the first one under the qualitative assumption that the function to be estimated belongs to some Hoelder class #SIGMA#(#beta#, L) with unknown parameters #beta#, L. This assumption allows to choose a family of kernel in an optimal way and the resulting procedure appears to be asymptotically optimal in the adaptive sense. (orig.)SIGLEAvailable from TIB Hannover: RR 5549(229)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Weierstrass-Institut fuer Angewandte Analysis und Stochastik (WIAS). Jahresforschungsbericht 1997

Available from TIB Hannover / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Numerical methods for kinetic equations Book of abstracts

SIGLEAvailable from TIB Hannover: F97B2212+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Geschichten der Thermodynamik und obskure Anwendungen des zweiten Hauptsatzes

Available from TIB Hannover: RR 5549(330)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Eigen mode solver for microwave transmission lines

The electromagnetic properties of microwave transmission lines can be described using Maxwell's equations in the frequency domain. Applying a finite-volume scheme this results in an algebraic eigenmode problem. In this paper, an improved numerical computation of the eigenmodes is presented. (orig.)SIGLEAvailable from TIB Hannover: RR 5549(308)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

A duality approach in the optimization of beams and plates

We introduce a class of nonlinear transformations called 'resizing rules' which associate to optimal shape design problems certain equivalent distributed control problems, while preserving the state of the system. This puts into evidence the duality principle that the class of system states that can be achieved, under a prescribed force, via modifications of the structure (shape) of the system can be as well obtained via the modifications of the force action, under a prescribed structure. We apply such transformations to the optimization of beams and plates and, in the simply supported or in the cantilevered cases, the obtained control problems are even convex. In all cases, we establish existence theorems for optimal pairs, by assuming only boundedness conditions. Moreover, in the simply supported case, we also prove the uniqueness of the global minimizer. A general algorithm that iterates between the original problem and the transformed one is introduced and studied. The applications also include the case of variational inequalities. (orig.)Available from TIB Hannover: RR 5549(335)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman