On a muted family of bivariate distributions

This paper introduces ‘muted bivariate distribution (MBD)’ as a new family of bivariate distributions possessing the property (P) that the concomitant of nth record value on one variable arising from any distribution of this family is distributed identically as the largest order statistic of a sample of size n arising from the marginal distribution of the same variable. Also if the property P is observed on any distribution, then we illustrate a process of impounding the probability density function (pdf) of the nth record value on one variable resulting due to P with the known form of the marginal pdf on the other variable to build up an appropriate MBD of the parent population. Some properties of the newly defined MBD are further derived. We identify the forms of the pdf of the concomitant of generalized upper record value and the pdf of the concomitant of the smallest order statistic of a sample of size n in order that each of the above pdf’s again can be used to generate MBD of the parent population, alternatively. We further observe MBD as a suitable model for describing an astronomical bivariate dataset relating to the galaxy luminosity measurements made under two conditions on globular clusters. For illustrative purposes, a real-life dataset on acacia trees obtained via concomitant record ranked set sampling methodology is considered as an application of this new family of bivariate distributions.     

A new approach to queer differential equations

We study the variational problem Where Ψ* is the increasing rearrangement of Ψ. An approximate problem is introduced which involves a variational problem with n free boundaries (n → ∞). Various estimates are established. In particular, when Ω is convex we show that the solution to the approximate problem is superharmonic and has bounded gradient.     

A class of bivariate distributions including the bivariate logistic

A univariate logistic distribution can be specified by considering a suitable form for the odds in favor of a failure against survival. This concept is extended to the bivariate case and a class of distributions, indexed by a parameter of association, having given marginals is proposed. Some properties of the proposed class of distributions are studied.     

A new approach to nonlinear partial differential equations

In this paper, a novel method called variational iteration method is proposed to solve nonlinear partial differential equations without linearization or small perturbations. In this method, a correction functional is constructed by a general Lagrange multiplier, which can be identified via variational theory. An analytical solution can be obtained from its trial-function with possible unknown constants, which can be identified by imposing the boundary conditions, by successively iteration.     

Joint distributions for stochastic functional differential equations

Consider stochastic functional differential equations, whose coefficients depend on past histories. The solution determines a non-Markov process. In the present paper, we shall obtain the existence of smooth densities for joint distributions of solutions, under the uniformly elliptic condition on the diffusion coefficients, via the Malliavin calculus. As an application, we shall study the computations of the Greeks on options associated with the asset price dynamics models with delayed effects.     

A new approach to nonlinear partial differential equations

The author's decomposition method for the solution of operator equations which may be nonlinear and/or stochastic is generalized to multidimensional cases.     

Sarmanov family of multivariate distributions for bivariate dynamic claim counts model

To predict future claims, it is well-known that the most recent claims are more predictive than older ones. However, classic panel data models for claim counts, such as the multivariate negative binomial distribution, do not put any time weight on past claims. More complex models can be used to consider this property, but often need numerical procedures to estimate parameters. When we want to add a dependence between different claim count types, the task would be even more difficult to handle. In this paper, we propose a bivariate dynamic model for claim counts, where past claims experience of a given claim type is used to better predict the other type of claims. This new bivariate dynamic distribution for claim counts is based on random effects that come from the Sarmanov family of multivariate distributions. To obtain a proper dynamic distribution based on this kind of bivariate priors, an approximation of the posterior distribution of the random effects is proposed. The resulting model can be seen as an extension of the dynamic heterogeneity model described in Bolancé et al. (2007). We apply this model to two samples of data from a major Canadian insurance company, where we show that the proposed model is one of the best models to adjust the data. We also show that the proposed model allows more flexibility in computing predictive premiums because closed-form expressions can be easily derived for the predictive distribution, the moments and the predictive moments.     

S-Bounded distributions,application to partial differential equations

The S-boundness at infinity of a distribution is defined to give some informations on the behaviour of a large class of distributions at infinity. The subspace A′ ? D′ of S-bounded distributions has been characterized and the properties of elements of A′ have been analyzed especially those interesting for partial differential equations. At the end, some propositions, concerning the S-boundness of solutions of linear partial differential equations have been proved.     

A novel approach to construct numerical methods for stochastic differential equations

In this paper we propose a new numerical method for solving stochastic differential equations (SDEs). As an application of this method we propose an explicit numerical scheme for a super linear SDE for which the usual Euler scheme diverges.     

A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations

In this paper, the pseudo-spectral method is generalized for solving fractional differential equations with initial conditions. For this purpose, an appropriate representation of the solution is presented and the pseudo-spectral differentiation matrix of fractional order is derived. Then, by using pseudo-spectral scheme, the problem is reduced to the solution of a system of algebraic equations. Through several numerical examples, we evaluate the accuracy and performance of our proposed method.     

On the period function for a family of complex differential equations

We consider planar differential equations of the form being f(z) and g(z) holomorphic functions and prove that if g(z) is not constant then for any continuum of period orbits the period function has at most one isolated critical period, which is a minimum. Among other implications, the paper extends a well-known result for meromorphic equations, that says that any continuum of periodic orbits has a constant period function.     

A self-dual variational approach to stochastic partial differential equations

Unlike many of their deterministic counterparts, stochastic partial differential equations are not amenable to the methods of calculus of variations à la Euler–Lagrange. In this paper, we show how self-dual variational calculus leads to variational solutions of various stochastic partial differential equations driven by monotone vector fields. We construct solutions as minima of suitable non-negative and self-dual energy functionals on Itô spaces of stochastic processes. We show how a stochastic version of Bolza's duality leads to solutions for equations with additive noise. We then use a Hamiltonian formulation to construct solutions for non-linear equations with non-additive noise such as the stochastic Navier–Stokes equations in dimension two.     

A new approach to monotone iterative technique for solution of differential equations

We discuss a monotone interative technique for initial value problems. The novelty of the approach is the use of g-monotonicity, i.e. of monotonicity of the ratio of two functions. This is in contrast to the more usual notion of monotonicity     

Persistence properties for a family of nonlinear partial differential equations

We examine the persistence of decay properties for a family of dispersive nonlinear partial differential equations. We show that certain decay properties of the initial data persist for as long as the solution exists. On the other hand, for a subset of the family certain decay rates are possible only for the trivial solution. For example, the only solution that remains with compact support for any further time is the trivial solution.     

A probabilistic approach to one class of nonlinear differential equations

Summary We establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes. As an application, we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations.Partially supported by National Science Foundation Grant DMS-8802667     

A class of bivariate exponential distributions

We introduce a class of absolutely continuous bivariate exponential distributions, generated from quadratic forms of standard multivariate normal variates.This class is quite flexible and tractable, since it is regulated by two parameters only, derived from the matrices of the quadratic forms: the correlation and the correlation of the squares of marginal components. A simple representation of the whole class is given in terms of 4-dimensional matrices. Integral forms allow evaluating the distribution function and the density function in most of the cases.The class is introduced as a subclass of bivariate distributions with chi-square marginals; bounds for the dimension of the generating normal variable are underlined in the general case.Finally, we sketch the extension to the multivariate case.     

A computer science approach for solving elliptic differential equations

A method that combines computer science and numerical analysis techniques for the solution of elliptic partial differential equations in domains consisting of the union of rectangles is presented. It is applied to Laplace's equation and to the Navier–Stokes equations in several domains. The method is very flexible and can be easily modified to solve other equations. © 1995 John Wiley & Sons, Inc.