Modeling Multivariate Interest Rates Using Time-Varying Copulas and Reducible Nonlinear Stochastic Differential Equations

We propose a new approach for modeling nonlinear multivariate interest rate processes based on time-varying copulas and reducible stochastic differential equations (SDEs). In the modeling of the marginal processes, we consider a class of nonlinear SDEs that are reducible to Ornstein--Uhlenbeck (OU) process or Cox, Ingersoll, and Ross (1985) (CIR) process. The reducibility is achieved via a nonlinear transformation function. The main advantage of this approach is that these SDEs can account for nonlinear features, observed in short-term interest rate series, while at the same time leading to exact discretization and closed-form likelihood functions. Although a rich set of specifications may be entertained, our exposition focuses on a couple of nonlinear constant elasticity volatility (CEV) processes, denoted as OU-CEV and CIR-CEV, respectively. These two processes encompass a number of existing models that have closed-form likelihood functions. The transition density, the conditional distribution function, and the steady-state density function are derived in closed form as well as the conditional and unconditional moments for both processes. In order to obtain a more flexible functional form over time, we allow the transformation function to be time varying. Results from our study of U.S. and UK short-term interest rates suggest that the new models outperform existing parametric models with closed-form likelihood functions. We also find the time-varying effects in the transformation functions statistically significant. To examine the joint behavior of interest rate series, we propose flexible nonlinear multivariate models by joining univariate nonlinear processes via appropriate copulas. We study the conditional dependence structure of the two rates using Patton (2006a) time-varying symmetrized Joe--Clayton copula. We find evidence of asymmetric dependence between the two rates, and that the level of dependence is positively related to the level of the two rates. (JEL: C13, C32, G12) Copyright The Author 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org, Oxford University Press.

Discretization and solution of the inhomogeneous Bogoliubov-de Gennes equations

In the presence of inhomogeneities, defects and currents, the equations describing a Bose-condensed ensemble of alkali atoms have to be solved numerically. By combining both linear and nonlinear equations within a Discrete Variable Representation framework, we describe a computational scheme for the solution of the coupled Bogoliubov-de Gennes (BdG) and nonlinear Schrodinger (NLS) equations for fields in a 3D spheroidal potential. We use the method to calculate the collective excitation spectrum and quasiparticle mode densities for excitations of a Bose condensed gas in a spheroidal trap. The method is compared against finite-difference and spectral methods, and we find the DVR computational scheme to be superior in accuracy and efficiency for the cases we consider. (C) 2004 Elsevier B.V. All rights reserved.

A review of the equations used to predict the velocity distribution within a ship’s propeller jet

Predicting the velocity within the ship’s propeller jet is the initial step to investigate the scouring made by the propeller jet. Albertson et al. (1950) suggested the investigation of a submerged jet can be undertaken through observation of the plain water jet from an orifice. The plain water jet investigation of Albertson et al. (1950) was based on the axial momentum theory. This has been the basis of all subsequent work with propeller jets. In reality, the velocity characteristic of a ship’s propeller jet is more complicated than a plain water jet. Fuehrer and Römisch (1977), Blaauw and van de Kaa (1978), Berger et al. (1981), Verhey (1983) and Hamill (1987) have carried out investigations using physical model. This paper reviews the state-of-art of the equations used to predict the time-averaged axial, tangential and radial components of velocity within the zone of flow establishment and the zone of established flow of a ship’s propeller jet.

Critical assessment of two-qubit post-Markovian master equations

A post-Markovian master equation has been recently proposed as a tool to describe the evolution of a system coupled to a memory-keeping environment [A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101 ( R) ( 2005)]. For a single qubit affected by appropriately chosen environmental conditions, the corresponding dynamics is always legitimate and physical. Here we extend such a situation to the case of two qubits, only one of which experiences the environmental effects. We show how, despite the innocence of such an extension, the introduction of the second qubit should be done cum grano salis to avoid consequences such as the breaking of the positivity of the associated dynamical map. This hints at the necessity of using care when adopting phenomenologically derived models for evolutions occurring outside the Markovian framework.

Cold-formed steel sections with web openings subjected to web crippling under two-flange loading conditions - Part II: Parametric study and proposed design equations

A parametric study of cold-formed steel sections with web openings subjected to web crippling was undertaken using finite element analysis, to investigate the effects of web holes and cross-section sizes on the web crippling strengths of channel sections subjected to web crippling under both interior-two-flange (ITF) and end-two-flange (ETF) loading conditions. In both loading conditions, the hole was centred beneath the bearing plate. It was demonstrated that the main factors influencing the web crippling strength are the ratio of the hole depth to the flat depth of the web, and the ratio of the length of bearing plates to the flat depth of the web. In this paper, design recommendations in the form of web crippling strength reduction factors are proposed, that are conservative to both the experimental and finite element results.

Two non-synonymous markers in PTPN21, identified by genome-wide association study data-mining and replication, are associated with schizophrenia

We conducted data-mining analyses of genome wide association (GWA) studies of the CATIE and MGS-GAIN datasets, and found 13 markers in the two physically linked genes, PTPN21 and EML5, showing nominally significant association with schizophrenia. Linkage disequilibrium (LD) analysis indicated that all 7 markers from PTPN21 shared high LD (r(2)>0.8), including rs2274736 and rs2401751, the two non-synonymous markers with the most significant association signals (rs2401751, P=1.10 × 10(-3) and rs2274736, P=1.21 × 10(-3)). In a meta-analysis of all 13 replication datasets with a total of 13,940 subjects, we found that the two non-synonymous markers are significantly associated with schizophrenia (rs2274736, OR=0.92, 95% CI: 0.86-0.97, P=5.45 × 10(-3) and rs2401751, OR=0.92, 95% CI: 0.86-0.97, P=5.29 × 10(-3)). One SNP (rs7147796) in EML5 is also significantly associated with the disease (OR=1.08, 95% CI: 1.02-1.14, P=6.43 × 10(-3)). These 3 markers remain significant after Bonferroni correction. Furthermore, haplotype conditioned analyses indicated that the association signals observed between rs2274736/rs2401751 and rs7147796 are statistically independent. Given the results that 2 non-synonymous markers in PTPN21 are associated with schizophrenia, further investigation of this locus is warranted.

R-matrix-Floquet theory of multiphoton processes:VIII. A linear equations method

A new linear equations method for calculating the R-matrix, which arises in the R-matrix-Floquet theory of multiphoton processes, is introduced. This method replaces the diagonalization of the Floquet Hamiltonian matrix by the solution of a set of linear simultaneous equations which are solved, in the present work, by the conjugate gradient method. This approach uses considerably less computer memory and can be readily ported onto parallel computers. It will thus enable much larger problems of current interest to be treated. This new method is tested by applying it to three-photon ionization of helium at frequencies where double resonances with a bound state and autoionizing states are important. Finally, an alternative linear equations method, which avoids the explicit calculation of the R-matrix by incorporating the boundary conditions directly, is described in an appendix.

Sound Synthesis for Contact-Driven Musical Instruments via Discretisation of Hamilton's Equations

Physical modelling of musical instruments involves studying nonlinear interactions between parts of the instrument. These can pose several difficulties concerning the accuracy and stability of numerical algorithms. In particular, when the underlying forces are non-analytic functions of the phase-space variables, a stability proof can only be obtained in limited cases. An approach has been recently presented by the authors, leading to unconditionally stable simulations for lumped collision models. In that study, discretisation of Hamilton’s equations instead of the usual Newton’s equation of motion yields a numerical scheme that can be proven to be energy conserving. In this paper, the above approach is extended to collisions of distributed objects. Namely, the interaction of an ideal string with a flat barrier is considered. The problem is formulated within the Hamiltonian framework and subsequently discretised. The resulting nonlinearmatrix equation can be shown to possess a unique solution, that enables the update of the algorithm. Energy conservation and thus numerical stability follows in a way similar to the lumped collision model. The existence of an analytic description of this interaction allows the validation of the model’s accuracy. The proposed methodology can be used in sound synthesis applications involving musical instruments where collisions occur either in a confined (e.g. hammer-string interaction, mallet impact) or in a distributed region (e.g. string-bridge or reed-mouthpiece interaction).

Scaling of magneto-quantum-radiative hydrodynamic equations: From laser-produced plasmas to astrophysics

We introduce the equations of magneto-quantum-radiative hydrodynamics. By rewriting them in a dimensionless form, we obtain a set of parameters that describe scale-dependent ratios of characteristic hydrodynamic quantities. We discuss how these dimensionless parameters relate to the scaling between astrophysical observations and laboratory experiments.

Nonequilibrium processes from generalized Langevin equations: Realistic nanoscale systems connected to two thermal baths

We extend the generalized Langevin equation (GLE) method [L. Stella, C. D. Lorenz, and L. Kantorovich, Phys. Rev. B 89, 134303 (2014)] to model a central classical region connected to two realistic thermal baths at two different temperatures. In such nonequilibrium conditions a heat flow is established, via the central system, in between the two baths. The GLE-2B (GLE two baths) scheme permits us to have a realistic description of both the dissipative central system and its surrounding baths. Following the original GLE approach, the extended Langevin dynamics scheme is modified to take into account two sets of auxiliary degrees of freedom corresponding to the mapping of the vibrational properties of each bath. These auxiliary variables are then used to solve the non-Markovian dissipative dynamics of the central region. The resulting algorithm is used to study a model of a short Al nanowire connected to two baths. The results of the simulations using the GLE-2B approach are compared to the results of other simulations that were carried out using standard thermostatting approaches (based on Markovian Langevin and Nosé-Hoover thermostats). We concentrate on the steady-state regime and study the establishment of a local temperature profile within the system. The conditions for obtaining a flat profile or a temperature gradient are examined in detail, in agreement with earlier studies. The results show that the GLE-2B approach is able to treat, within a single scheme, two widely different thermal transport regimes, i.e., ballistic systems, with no temperature gradient, and diffusive systems with a temperature gradient.

Exact and numerical solutions for diffusio-reaction equations with convection term

Esta dissertação estuda essencialmente dois problemas: (A) uma classe de equações unidimensionais de reacção-difusão-convecção em meios não uniformes (dependentes do espaço), e (B) um problema elíptico não-linear e paramétrico ligado a fenómenos de capilaridade. A Análise de Perturbação Singular e a dinâmica de Hamilton-Jacobi são utilizadas na obtenção de expressões assimptóticas para a solução (com comportamento de frente) e para a sua velocidade de propagação. Os seguintes três métodos de decomposição, Adomian Decomposition Method (ADM), Decomposition Method based on Infinite Products (DIP), e New Iterative Method (NIM), são apresentados e brevemente comparados. Adicionalmente, condições suficientes para a convergência da solução em série, obtida pelo ADM, e uma aplicação a um problema da Telecomunicações por Fibras Ópticas, envolvendo EDOs não-lineares designadas equações de Raman, são discutidas. Um ponto de vista mais abrangente que unifica os métodos de decomposição referidos é também apresentado. Para subclasses desta EDP são obtidas soluções numa forma explícita, para diferentes tipos de dados e usando uma variante do método de simetrias de Bluman-Cole. Usando Teoria de Pontos Críticos (o teorema usualmente designado mountain pass) e técnicas de truncatura, prova-se a existência de duas soluções não triviais (uma positiva e uma negativa) para o problema elíptico não-linear e paramétrico (B). A existência de uma terceira solução não trivial é demonstrada usando Grupos Críticos e Teoria de Morse.

Existence results for elliptic equations with singular terms

Esta dissertação estuda em detalhe três problemas elípticos: (I) uma classe de equações que envolve o operador Laplaciano, um termo singular e nãolinearidade com o exponente crítico de Sobolev, (II) uma classe de equações com singularidade dupla, o expoente crítico de Hardy-Sobolev e um termo côncavo e (III) uma classe de equações em forma divergente, que envolve um termo singular, um operador do tipo Leray-Lions, e uma função definida nos espaços de Lorentz. As não-linearidades consideradas nos problemas (I) e (II), apresentam dificuldades adicionais, tais como uma singularidade forte no ponto zero (de modo que um "blow-up" pode ocorrer) e a falta de compacidade, devido à presença do exponente crítico de Sobolev (problema (I)) e Hardy-Sobolev (problema (II)). Pela singularidade existente no problema (III), a definição padrão de solução fraca pode não fazer sentido, por isso, é introduzida uma noção especial de solução fraca em subconjuntos abertos do domínio. Métodos variacionais e técnicas da Teoria de Pontos Críticos são usados para provar a existência de soluções nos dois primeiros problemas. No problema (I), são usadas uma combinação adequada de técnicas de Nehari, o princípio variacional de Ekeland, métodos de minimax, um argumento de translação e estimativas integrais do nível de energia. Neste caso, demonstramos a existência de (pelo menos) quatro soluções não triviais onde pelo menos uma delas muda de sinal. No problema (II), usando o método de concentração de compacidade e o teorema de passagem de montanha, demostramos a existência de pelo menos duas soluções positivas e pelo menos um par de soluções com mudança de sinal. A abordagem do problema (III) combina um resultado de surjectividade para operadores monótonos, coercivos e radialmente contínuos com propriedades especiais do operador de tipo Leray- Lions. Demonstramos assim a existência de pelo menos, uma solução no espaço de Lorentz e obtemos uma estimativa para esta solução.