An RBF scheme for option pricing in exponential Levy models

We use Radial Basis Function (RBF) interpolation to price options in exponential Lévy models by numerically solving the fundamental pricing PIDE (Partial integro-differential equations). Our RBF scheme can handle arbitrary singularities of the Lévy measure in 0 without introducing further approximations, making it simpler to implement than competing methods. In numerical experiments using processes from the CGMY-KoBoL class, the scheme is found to be second order convergent in the number of interpolation points, including for processes of unbounded variation

H2 molecule in strong magnetic fields

The Pauli-Hamiltonian of a molecule with fixed nuclei in a strong constant magnetic field is asymptotic, in norm-resolvent sense, to an effective Hamiltonian which has the form of a multi-particle Schr\"odinger operator with interactions given by one-dimensional \delta-potentials. We study this effective Hamiltonian in the case of the H2 -molecule and establish existence of the ground state. We also show that the inter-nuclear equilibrium distance tends to 0 as the field-strength tends to infinity

Convergence of Stationary RBF-schemes for the numerical solution of evolution equations

In this paper we establish convergence rates for semi-discrete stationary RBF schemes for the classical heat equation and, more generally, for a large class of translation invariant pseudo-differenti al evolution equations which include the fractional heat equation and the Kolmogorov-Fokker-Planck equations of Levy processes (under natural conditions on the Levy measure), but also hyperbolic equations such as the half-wave equation

Auto-tail dependence coefficients for stationary solutions of linear stochastic recurrence equations and for GARCH(1,1)

We examine the auto-dependence structure of strictly stationary solutions of linear stochastic recurrence equations and of strictly stationary GARCH(1, 1) processes from the point of view of ordinary and generalized tail dependence coefficients. Since such processes can easily be of infinite variance, a substitute for the usual auto-correlation function is needed

Functionals of exponential Brownian motion and divided differences

We provide a surprising new application of classical approximation theory to a fundamental asset-pricing model of mathematical finance. Specifically, we calculate an analytic value for the correlation coefficient between exponential Brownian motion and its time average, and we find the use of divided differences greatly elucidates formulae, providing a path to several new results. As applications, we find that this correlation coefficient is always at least 1/p2 and, via the Hermite–Genocchi integral relation, demonstrate that all moments of the time average are certain divided differences of the exponential function. We also prove that these moments agree with the somewhat more complex formulae obtained by Oshanin and Yor

Teaching and learning for the future

This is the final report of the Committee on MultiMedia in Teacher Training (COMMITT), which offers a strategic framework to support efforts of teacher training institutes in the Netherlands to develop their own plans for enhancing the teaching and learning process as well as its outcomes through the application of Information and Communication Technology (ICT). The purpose of the plan of action, called "a leap for the future," is to serve as a catalyst for teacher training institutes, elementary and secondary schools, and universities to work together on a common goal: improving teaching and learning through the application of ICT. This report consists of seven chapters. The first chapter is an introduction to the initiative to revitalize teacher training and the committee's activities. Chapter 2 describes an analysis of the transformation of society and consequences for learning in the future. Chapter 3 discusses key elements in the teaching and learning process. A conceptual model is presented which serves as a frame of reference for COMMITT. The role of ICT in teaching and learning is also discussed. Chapter 4 outlines the scope of COMMITT and provides examples illustrating the impact of ICT as a medium for education. Chapter 5 describes the challenges, opportunities and possible threats of implementing ICT in the education system and teacher training institutes. Guidelines for a program of action are elaborated in Chapter 6. The role of government, vision underlying the program, strategy and program of action, management and organization, and budget are discussed. Chapter 7 includes concluding remarks with a special emphasis on issues and factors which are expected to influence the implementation of the "leap for the future." Appendices present a list of the COMMITT members; a discussion on the Dutch education system; Committee statements; statements developed and judged by ICT coordinators; and different types of use of ICT as a medium. (AEF

On the maximal ionization of atoms in strong magnetic fields

We give upper bounds for the number of spin 1/2 particles that can be bound to a nucleus of charge Z in the presence of a magnetic field B, including the spin-field coupling. We use Lieb's strategy, which is known to yield N_c<2Z+1 for magnetic fields that go to zero at infinity, ignoring the spin-field interaction. For particles with fermionic statistics in a homogeneous magnetic field our upper bound has an additional term of order $Z\times\min{(B/Z^3)^{2/5},1+|\ln(B/Z^3)|^2}$.Comment: LaTeX2e, 8 page

Spectral fluctuations of Schr\"odinger operators generated by critical points of the potential

Starting from the spectrum of Schr\"odinger operators on $\mathbb{R}^n$, we propose a method to detect critical points of the potential. We argue semi-classically on the basis of a mathematically rigorous version of Gutzwiller's trace formula which expresses spectral statistics in term of classical orbits. A critical point of the potential with zero momentum is an equilibrium of the flow and generates certain singularities in the spectrum. Via sharp spectral estimates, this fluctuation indicates the presence of a critical point and allows to reconstruct partially the local shape of the potential. Some generalizations of this approach are also proposed.\medskip keywords : Semi-classical analysis; Schr\"odinger operators; Equilibriums in classical mechanics.Comment: 18 pages, Final versio

Auto-dependence structure of arch-models: tail dependence coefficients

We study autodependence in ARCH-models by computing the auto-lower tail dependence coefficients and certain generalizations thereof, for both stationary and non-stationary time series. This study is inspired by financial risk-management issues, and our results are relevant for estimating probabilities of consecutive value-at-risk violations