Global existence and compact attractors for the discrete nonlinear Schrödinger equation

AbstractWe study the asymptotic behavior of solutions of discrete nonlinear Schrödinger-type (DNLS) equations. For a conservative system, we consider the global in time solvability and the question of existence of standing wave solutions. Similarities and differences with the continuous counterpart (NLS-partial differential equation) are pointed out. For a dissipative system we prove existence of a global attractor and its stability under finite-dimensional approximations. Similar questions are treated in a weighted phase space. Finally, we propose possible extensions for various types of DNLS equations

Probability of Default modelling with L\'evy-driven Ornstein-Uhlenbeck processes and applications in credit risk under the IFRS 9

In this paper we develop a framework for estimating Probability of Default (PD) based on stochastic models governing an appropriate asset value processes. In particular, we build upon a L\'evy-driven Ornstein-Uhlenbeck process and consider a generalized model that incorporates multiple latent variables affecting the evolution of the process. We obtain an Integral Equation (IE) formulation for the corresponding PD as a function of the initial position of the asset value process and the time until maturity, from which we then prove that the PD function satisfies an appropriate Partial Integro-Differential Equation (PIDE). These representations allow us to show that appropriate weak (viscosity) as well as strong solutions exist, and develop subsequent numerical schemes for the estimation of the PD function. Such a framework is necessary under the newly introduced International Financial Reporting Standards (IFRS) 9 regulation, which has imposed further requirements on the sophistication and rigor underlying credit modelling methodologies. We consider special cases of the generalized model that can be used for applications to credit risk modelling and provide examples specific to provisioning under IFRS 9, and more

Diffusion models in strongly chaotic Hamiltonian systems

The main subject of this thesis is the long time behaviour of strongly chaotic Hamiltonian systems and whether their behaviour ran be modelled with diffusion processes. The problem of diffusion caused by chaos in a particular area preserving map on the torus, the web map is studied. The formalism is then generalised for the study of diffusion in higher dimensional symplectic maps on the cylinder and general results are obtained. A numerical method for the calculation of diffusion coefficients for chaotic maps is described. Finally, the problem of diffusion in phase space in the case where chaos coexists with structures such as stable islands is studied

Statistical monitoring of functional data using the notion of Fr\'echet mean combined with the framework of the deformation models

The aim of this paper is to investigate possible advances obtained by the implementation of the framework of Fr\'echet mean and the generalized sense of mean that it offers, in the field of statistical process monitoring and control. In particular, the case of non-linear profiles which are described by data in functional form is considered and a framework combining the notion of Fr\'echet mean and deformation models is developed. The proposed monitoring approach is implemented to the intra-day air pollution monitoring task in the city of Athens where the capabilities and advantages of the method are illustrated.Comment: 31 pages, 11 figure

Electromagnetic fields in linear and nonlinear chiral media: a time-domain analysis

We present several recent and novel results on the formulation and the analysis of the equations governing the evolution of electromagnetic fields in chiral media in the time domain. In particular, we present results concerning the well-posedness and the solvability of the problem for linear, time-dependent, and nonlocal media, andresults concerning the validity of the local approximation of the nonlocal medium (optical response approximation). The paper concludes with the study of a class of nonlinear chiral media exhibiting Kerr-like nonlinearities, for which the existence of bright and dark solitary waves is shown