Heat Equation on the Cone and the Spectrum of the Spherical Laplacian

Spectrum of the Laplacian on spherical domains is analyzed from the point of view of the heat equation on the cone. The series solution to the heat equation on the cone is known to lead to a study of the Laplacian eigenvalue problem on domains on the sphere in higher dimensions. It is found that the solution leads naturally to a spectral function, a `generating function' for the eigenvalues and multiplicities of the Laplacian, expressible in closed form for certain domains on the sphere. Analytical properties of the spectral function suggest a simple scaling procedure for estimating the eigenvalues. Comparison of the first eigenvalue estimate with the available theoretical and numerical results for some specific domains shows remarkable agreement.Comment: 16 page

Levy subordinator model: A two parameter model of default dependency

The May 2005 crisis and the recent credit crisis have indicated to us that any realistic model of default dependency needs to account for at least two risk factors, firm-specific and catastrophic. Unfortunately, the popular Gaussian copula model has no identifiable support to either of these. In this article, a two parameter model of default dependency based on the Levy subordinator is presented accounting for these two risk factors. Subordinators are Levy processes with non-decreasing sample paths. They help ensure that the loss process is non-decreasing leading to a promising class of dynamic models. The simplest subordinator is the Levy subordinator, a maximally skewed stable process with index of stability 1/2. Interestingly, this simplest subordinator turns out to be the appropriate choice as the basic process in modeling default dependency. Its attractive feature is that it admits a closed form expression for its distribution function. This helps in automatic calibration to individual hazard rate curves and efficient pricing with Fast Fourier Transform techniques. It is structured similar to the one-factor Gaussian copula model and can easily be implemented within the framework of the existing infrastructure. As it turns out, the Gaussian copula model can itself be recast into this framework highlighting its limitations. The model can also be investigated numerically with a Monte Carlo simulation algorithm. It admits a tractable framework of random recovery. It is investigated numerically and the implied base correlations are presented over a wide range of its parameters. The investigation also demonstrates its ability to generate reasonable hedge ratios

Levy Density Based Intensity Modeling of the Correlation Smile

The jump distribution for the default intensities in a reduced form framework is modeled and calibrated to provide reasonable fits to CDX.NA.IG and iTraxx Europe CDOs, to 5, 7 and 10 year maturities simultaneously. Calibration is carried out using an efficient Monte Carlo simulation algorithm suitable for both homogeneous and heterogeneous collections of credit names. The underlying jump process is found to relate closely to a maximally skewed stable Levy process with index of stability alpha ~ 1.5.Default Risk; Default Correlation; Default Intensity; Intensity Model; Levy Density; CDO; Monte Carlo

A Semi-Analytical Parametric Model for Dependent Defaults

A semi-analytical parametric approach to modeling default dependency is presented. It is a multi-factor model based on instantaneous default correlation that also takes into account higher order default correlations. It is capable of accommodating a term structure of default correlations and has a dynamic formulation in the form of a continuous time Markov chain. With two factors and a constant hazard rate, it provides perfect fits to four tranches of CDX.NA.IG and iTraxx Europe CDOs of 5, 7 and 10 year maturities. With time dependent hazard rates, it provides perfect fits to all the five tranches for all three maturities.Default Risk; Default Correlation; CDO; Markov Chain; Semi-analytical; Parametric

Delayed Default Dependency and Default Contagion

Delayed, hence non-simultaneous, dependent defaults are discussed in a reduced form model. The model is a generalization of a multi-factor model based on simultaneous defaults to incorporate delayed defaults. It provides a natural smoothening of discontinuities in the joint probability densities in models with simultaneous defaults. It is a dynamic model that exhibits default contagion in a multi-factor setting. It admits an efficient Monte Carlo simulation algorithm that can handle heterogeneous collections of credit names. It can be calibrated to provide exact fits to CDX.NA.IG and iTraxx Europe CDOs just as its version with simultaneous defaults.Default Risk; Default Correlation; Default Contagion; Delayed Default; CDO; Monte Carlo

From Cosmic Inflation and Matter Creation to Dark Matter -- Journey of the Inflaton?

A scenario of the inflaton evolution from cosmic inflation and matter creation to dark matter/dark energy today is presented. To start with, a model of the inflationary phase of the inflaton is introduced. The inflaton rolls down a simple quadratic hilltop potential along with matter creation being dragged down by the presence of matter. Presence of matter provides a mechanism to stop universe's acceleration and hence the inflationary phase. The model predictions for the standard metrics are fully consistent with the current CMB limits. The quadratic potential could in principle be extended to complete a potential hill subsequent to inflation. The evolution of the inflaton from the inflationary phase to radiation/matter dominated eras and to current times can be inferred qualitatively following the evolution of its equation of state parameter. The existence of solutions to its dynamics, tracking matter as it evolves to current times, provides a plausible reasoning for the relative order of magnitudes of the cosmological parameters, in particular to the relative abundance of dark matter today.Comment: arXiv admin note: text overlap with arXiv:2210.1547

1mb{1\over m_b} and 1mt{1\over m_t} Expansion of the Weak Mixing Matrix

We perform a $1/m_b$ and $1/m_t$ expansion of the Cabibbo-Kobayashi- Maskawa mixing matrix. Data suggest that the dominant parts of the Yukawa couplings are factorizable into sets of numbers $\vert r>$, $\vert s>$, and $\vert s'>$, associated, respectively, with the left-handed doublets, the right-handed up singlets, and the right- handed down singlets. The first order expansion is consistent with Wolfenstein parameterization, which is an expansion in $sin \theta _c$ to third order. The mixing matrix elements in the present approach are partitioned into factors determined by the relative orientations of $\vert r>$, $\vert s>$, and $\vert s'>$ and the dynamics provided by the subdominant mass matrices. A short discussion is given of some experimental support and a generalized Fritzsch model is used to contrast our approach.Comment: A set of references has been added to ealier related wor

Levy Subordinator Model of Default Dependency

This article presents a model of default dependency based on Levy subordinator. It is a tractable dynamical model, computationally structured similar to the one-factor Gaussian copula model, providing easy calibration to individual hazard rate curves and efficient pricing with Fast Fourier Transform techniques. The subordinator is an alpha=1/2 stable Levy process, maximally skewed to the right, with its distribution function known in closed form as the Levy distribution. The model provides a reasonable fit to market data with just two parameters to assess dependency risk, a measure of correlation and that of the likelihood of a catastrophe