A converse of asymptotic formulae in simultaneous approximation

In the general setting of simultaneous approximation by sequences of linear shape preserving operators, this paper contains a sort of converse result of Voronovskaya-type asymptotic formulae. As a by-product a saturation result is derived. Applications to some very well-known approximation processes are also presented.     

A note on asymptotic formulae for one-dimensional network flow problems

This note develops asymptotic formulae for single-commodity network flow problems with random inputs. The transportation linear programming problem (TLP) where N points lie in a region of R¹ is one example. It is found that the average distance traveled by an item in the TLP increases with N^1/2; i.e., the unit cost is unbounded when N and the length of the region are increased in a fixed ratio. Further, the optimum distance does not converge in probability to the average value. These one-dimensional results are a useful stepping stone toward a network theory for two and higher dimensions. Research supported in part by the University of California Transportation Center.     

Accuracy of staircase approximations in finite-difference methods for wave propagation

While a number of increasingly sophisticated numerical methods have been developed for time-dependent problems in electromagnetics, the Yee scheme is still widely used in the applied fields, mainly due to its simplicity and computational efficiency. A fundamental drawback of the method is the use of staircase boundary approximations, giving inconsistent results. Usually experience of numerical experiments provides guidance of the impact of these errors on the final simulation result. In this paper, we derive exact discrete solutions to the Yee scheme close to the staircase approximated boundary, enabling a detailed theoretical study of the amplitude, phase and frequency errors created. Furthermore, we show how evanescent waves of amplitude \(O(1)\) occur along the boundary. These characterize the inconsistencies observed in electromagnetic simulations and the locality of the waves explain why, in practice, the Yee scheme works as well as it does. The analysis is supported by detailed proofs and numerical examples.     

A simplified analytical approach for pricing discretely-sampled variance swaps with stochastic volatility

Pricing variance swaps under stochastic volatility with discretely-sampled realized variance has been a hot subject pursued recently; quite a few papers have already been published (Zhu and Lian (2009, 2011, [11], [4]); Swishchuk and Li (2011) [5]). In this paper, we present a simplified approach to price discretely-sampled variance swaps. Compared with the approach presented by Zhu and Lian (2011) [4], an important feature of our approach is that there is no need for the introduction of a new state variable and the utilization of the generalized Fourier transform. This has significantly simplified the solution procedure and will thus enable researchers to view this type of problems from a different angle.     

Error bounds for asymptotic approximations of the partition function

We consider Markovian queueing models with a finite number of states and a product form solution for its steady state probability distribution. Starting from the integral representation for the partition function in complex space we construct error bounds for its asymptotic expansion obtained by the saddle point method. The derivation of error bounds is based on an idea by Olver applicable to integral transforms with an exponentially decaying kernel. The bounds are expressed in terms of the supremum of a certain function and are asymptotic to the absolute value of the first neglected term in the expansion as the large parameter approaches infinity. The application of these error bounds is illustrated for two classes of queueing models: loss systems and single chain closed queueing networks.     

Determinantal formulae for matrices with sparse inverses,II: asymmetric zero patterns

In an earlier paper, formulae for det A as a ratio of products of principal minors of A were exhibited, for any given symmetric zero-pattern of A⁻¹. These formulae may be presented in terms of a spanning tree of the intersection graph of certain index sets associated with the zero pattern of A⁻¹. However, just as the determinant of a diagonal and of a triangular matrix are both the product of the diagonal entries, the symmetry of the zero pattern is not essential for these formulae. We describe here how analogous formulae for det A may be obtained in the asymmetric-zero-pattern case by introducing a directed spanning tree. We also examine the converse question of determining all possible zero patterns of A⁻¹ which guarantee that a certain determinantal formula holds.     

A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations

After studying Gaussian type quadrature formulae with mixed boundary conditions, we suggest a fast algorithm for computing their nodes and weights. It is shown that the latter are computed in the same manner as in the theory of the classical Gauss quadrature formulae. In fact, all nodes and weights are again computed as eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. Hence, we can adapt existing procedures for generating such quadrature formulae. Comparative results with various methods now in use are given. In the second part of this paper, new algorithms for spectral approximations for second-order elliptic problems are derived. The key to the efficiency of our algorithms is to find an appropriate spectral approximation by using the most accurate quadrature formula, which takes the boundary conditions into account in such a way that the resulting discrete system has a diagonal mass matrix. Hence, our algorithms can be used to introduce explicit resolutions for the time-dependent problems. This is the so-called lumped mass method. The performance of the approach is illustrated with several numerical examples in one and two space dimensions.

     

The pricing of perpetual convertible bond with credit risk

Convertible bond gives holder the right to choose a conversion strategy to maximize the bond value, and issuer also has the right to minimize the bond value in order to maximize equity value. When there is default occurring, conversion and calling strategies are invalid. In the framework of reduced form model, we reduce the price of convertible bond to variational inequalities, and the coefficients of variational inequalities are unbounded at the original point. Then the existence and uniqueness of variational inequality are proven. Finally, we prove that the conversion area, the calling area and the holding area are connected subsets of the state space.     

On the construction of asymptotic approximations for a nonautonomous wave equation

For a nonautonomous wave equation with homogeneous boundary conditions, we construct one-frequency approximations of asymptotic solutions by using periodic Ateb-functions. Resonance and nonresonance cases are considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 12, pp. 1714–1716, December, 1995.     

The asymptotic number of (0,1)-matrices with zero permanent

It is shown that Z_n/n·2^{n2 - n + 1}→1, where Z_n is the number of n × n (0,1)-matrices with zero permanent.     

Existence of Gompertz parameters with mortality deceleration rate and their asymptotic formulae for a large population

In this paper we provide an interval of existence of mortality rate parameters A and α, and their asymptotic expressions in a Gompertz survival model with mortality deceleration rate, in the absence of age specific mortality data.     

Another approach to asymptotic expansions for Euler’s approximations of semigroups

In [2], optimal bounds for the remainder terms in asymptotic expansions for Euler’s approximations of semigroups were derived. The approach was based on applications of the Fourier-Laplace transforms, which allowed one to reduce the problem to estimation of error terms in the Law of Large Numbers. In this paper, we propose an alternative (direct) approach based on application of certain integro-differential identities (the so-called multiplicative representations of differences). Such identities were introduced by Bentkus [3] and applied (see Bentkus and Paulauskas [4]) to derive the optimal convergence rates in Chernoff-type lemmas and Euler’s approximations of semigroups. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 267–284, April–June, 2006.     

Closer asymptotic approximations for the distributions of the power divergence goodness-of-fit statistics

Summary The members of the power divergence family of statistics all have an asymptotically equivalent χ² distribution (Cressie and Read [1]). An asymptotic expansion for the distribution function is derived which shows that the speed of convergence to this asymptotic limit is dependent on λ. Known results for Pearson'sX ² statistic and the log-likelihood ratio statistic then appear as special cases in a continuum rather than as separate (unrelated) expansions.     

The asymptotic zero distribution of multiple orthogonal polynomials associated with Macdonald functions

We study the asymptotic zero distribution of type II multiple orthogonal polynomials associated with two Macdonald functions (modified Bessel functions of the second kind). On the basis of the four-term recurrence relation, it is shown that, after proper scaling, the sequence of normalized zero counting measures converges weakly to the first component of a vector of two measures which satisfies a vector equilibrium problem with two external fields. We also give the explicit formula for the equilibrium vector in terms of solutions of an algebraic equation.     

A comparison of variational approximations for fast inference in mixed logit models

Variational Bayesian methods aim to address some of the weaknesses (computation time, storage costs and convergence monitoring) of mainstream Markov chain Monte Carlo based inference at the cost of a biased but more tractable approximation to the posterior distribution. We investigate the performance of variational approximations in the context of the mixed logit model, which is one of the most used models for discrete choice data. A typical treatment using the variational Bayesian methodology is hindered by the fact that the expectation of the so called log-sum-exponential function has no explicit expression. Therefore additional approximations are required to maintain tractability. In this paper we compare seven different possible bounds or approximations. We found that quadratic bounds are not sufficiently accurate. A recently proposed non-quadratic bound did perform well. We also found that the Taylor series approximation used in a previous study of variational Bayes for mixed logit models is only accurate for specific settings. Our proposed approximation based on quasi Monte Carlo sampling performed consistently well across all simulation settings while remaining computationally tractable.