A Method for Incorporating Transcendentally Small Terms into the Method of Matched Asymptotic Expansions

The essential ideas behind a method for incorporating exponentially small terms into the method of matched asymptotic expansions are demonstrated using an Ackerberg–O'Malley resonance problem and a spurious solutions problem of Carrier and Pearson. One begins with the application of the standard method of matched asymptotic expansions to obtain at least the leading terms in outer and inner (Poincaré-type) expansions; some, although not all, matching can be carried out at this stage. This is followed by the introduction of supplementary expansions whose gauge functions are transcendentally small compared to those in the standard expansions. Analysis of terms in these expansions allows the matching to be completed. Furthermore, the method allows for the inclusion of globally valid transcendentally small contributions to the asymptotic solution; it is well known that such terms may be numerically significant.     

Asymptotic Expansions in the Compound Poisson Limit Theorem

A triangular array of independent infinitesimal integer-valued random variables is considered. Asymptotic expansions for the probability distributions of sums of these variables are investigated in the case of the limiting compound Poisson laws.     

Complete Asymptotic Expansions for the Chlodovsky Polynomials

The purpose of this article is to study the local rate of convergence of the Chlodovsky operators (C_nf)(x). As the main results, we investigate their asymptotic behaviour and derive the complete asymptotic expansions of these operators. All the coefficients of n^?k (k = 1, 2,…) are calculated in terms of the Stirling numbers of first and second kind. We mention that analogous results for the Bernstein polynomials can be found in Lorentz [2 G. G. Lorentz ( 1953 ). Bernstein Polynomials . University of Toronto Press , Toronto . [Google Scholar]].     

有关调和数的更多渐近展开公式

基于指数型完全Bell多项式,建立了一个一般调和数渐近展开式,并给出展开式中系数的相应递推关系.由生成函数方法进一步推导出这些系数的具体表达式.另外,我们建立了两个在对数项里只含有奇数或偶数次幂项的lacunary调和数渐近展开式,     

A Matched Asymptotic Expansions Approach to Continuity Corrections for Discretely Sampled Options. Part 1: Barrier Options

We propose a general framework to model equity volatility for a firm financed by equity and additional non-equity sources of funds. The stochastic nature of equity volatility is endogenous, and comes from the impact of a change in the value of the firm's assets on the financial leverage. We first present the basic model, which is an extension of the Black-Scholes model, to value corporate securities. Second, we show for the first time in the option literature, that instantaneous equity volatility is a solution of a partial differential equation similar to Black-Scholes', although it is non-linear and in general does not have any analytical solution. However, analytical approximations for equity volatility are proposed for different capital structures: (1) equity and debt, (2) equity and warrants, and (3) equity, debt and warrants. They are shown to be very accurate.     

球面上热核的渐近展开

具体计算了球面Ｓｎ（１）（ｎ２）上热核的渐近展开式中前五项的系数，而根据已有的公式只能算出前四项．最后给出了展开式一般项的递推公式，发现它与Ｂｅｒｎｏｕｌｉ数有未曾想到的联系．根据不变量理论，我们可以确定任意ｎ维紧致无边Ｒｉｅｍａｎｎ流形上热核的渐近展开式中第五项的系数．     

A Matched Asymptotic Expansions Approach to Continuity Corrections for Discretely Sampled Options. Part 2: Bermudan Options

The paper discusses the ‘continuity correction’ that should be applied to connect the prices of discretely sampled American put options (i.e. Bermudan options) and their continuously‐sampled equivalents. Using a matched asymptotic expansions approach the correction is computed and related to that discussed by Broadie, Glasserman & Kou (1997 Broadie, M. 1997. A continuity correction for discretely sampled barrier options,. Mathematical Finance, 7: 325[Crossref], [Web of Science ®] , [Google Scholar]) (Mathematical Finance, 7, p.325 for barrier options. In the Bermudan case, the continuity correction is an order of magnitude smaller than in the corresponding barrier problem. It is also shown that the optimal exercise boundary in the discrete case is slightly higher than in the continuously sampled case.     

Asymptotic Expansions of Solutions to the Riccati Equation

Doklady Mathematics - Scalar real Riccati equations with coefficients expandable in convergent power series in a neighborhood of infinity are considered. Extendable solutions to equations of this...     

Non-uniform Estimates in Asymptotic Expansions with a Stable Limit Distribution

Under pseudomoment conditions a non-uniform bound is proved for the remainder in asymptotic expansions with a stable limit law, where the quality of the bound with respect to the number of random variables is as good as that with respect to the argument. Further, a theorem for large deviations in the symmetrical problem is obtained.     

Uniform Estimates of Remainders in Asymptotic Expansions of Solutions to the Problem on Eigenoscillations of a Piezoelectric Plate

A method for obtaining estimates of asymptotic remainders is presented. The constants in estimates are independent of the number of the eigenvalue, as well as of the small parameter h, the thickness of the plate. Owing to an information about connections between frequencies of eigenoscillations of the three-dimensional plates and its two-dimensional model obtained under various restrictions to h, it is possible to divide the asymptotics in collective and individual ones. Only in the case of the individual asymptotics, i.e., under rigid restrictions on h, it is possible to construct asymptotic expansions for the corresponding eigenvectors. We consider arbitrarily anizotropic composed cylindrical plates in whcih piezoeffects can dominate along longitudinal directions, as well as along transverse directions. The connectedness of elastic and electric fields Implies the appearance of a nontrivial dissipative components of the operator of the problem under consideration, but its spectrum remains real and positive. Bibliography: 43 titles.     

Asymptotic Expansions of Posterior Distributions in Nonregular Cases

We study the asymptotic behaviour of the posterior distributions for a one-parameter family of discontinuous densities. It is shown that a suitably centered and normalized posterior converges almost surely to an exponential limit in the total variation norm. Further, asymptotic expansions for the density, distribution function, moments and quantiles of the posterior are also obtained. It is to be noted that, in view of the results of Ghosh et al. (1994, Statistical Decision Theory and Related Topics V, 183-199, Springer, New York) and Ghosal et al. (1995, Ann. Statist., 23, 2145-2152), the nonregular cases considered here are essentially the only ones for which the posterior distributions converge. The results obtained here are also supported by a simulation experiment.     

Uniform Asymptotic Expansions for the Discrete Chebyshev Polynomials

The discrete Chebyshev polynomials t_n(x, N) are orthogonal with respect to a distribution function, which is a step function with jumps one unit at the points x = 0, 1, … , N ? 1, N being a fixed positive integer. By using a double integral representation, we derive two asymptotic expansions for t_n(aN, N + 1) in the double scaling limit, namely, N →∞ and n/N → b, where b ∈ (0, 1) and a ∈ (?∞, ∞). One expansion involves the confluent hypergeometric function and holds uniformly for , and the other involves the Gamma function and holds uniformly for a ∈ (?∞, 0). Both intervals of validity of these two expansions can be extended slightly to include a neighborhood of the origin. Asymptotic expansions for can be obtained via a symmetry relation of t_n(aN, N + 1) with respect to . Asymptotic formulas for small and large zeros of t_n(x, N + 1) are also given.     

Uniform Asymptotic Expansions of Symmetric Elliptic Integrals

Symmetric standard elliptic integrals are considered when two or more parameters are larger than the others. The distributional approach is used to derive seven expansions of these integrals in inverse powers of the asymptotic parameters. Some of these expansions also involve logarithmic terms in the asymptotic variables. These expansions are uniformly convergent when the asymptotic parameters are greater than the remaining ones. The coefficients of six of these expansions involve hypergeometric functions with less parameters than the original integrals. The coefficients of the seventh expansion again involve elliptic integrals, but with less parameters than the original integrals. The convergence speed of any of these expansions increases for an increasing difference between the asymptotic variables and the remaining ones. All the expansions are accompanied by an error bound at any order of the approximation. January 31, 2000. Date revised: May 18, 2000. Date accepted: August 4, 2000.     

Asymptotic Expansions of the Contact Angle in Nonlocal Capillarity Problems

We consider a family of nonlocal capillarity models, where surface tension is modeled by exploiting the family of fractional interaction kernels \(|z|^{-n-s}\), with \(s\in (0,1)\) and n the dimension of the ambient space. The fractional Young’s law (contact angle condition) predicted by these models coincides, in the limit as \(s\rightarrow 1^-\), with the classical Young’s law determined by the Gauss free energy. Here we refine this asymptotics by showing that, for s close to 1, the fractional contact angle is always smaller than its classical counterpart when the relative adhesion coefficient \(\sigma \) is negative, and larger if \(\sigma \) is positive. In addition, we address the asymptotics of the fractional Young’s law in the limit case \(s\rightarrow 0^+\) of interaction kernels with heavy tails. Interestingly, near \(s=0\), the dependence of the contact angle from the relative adhesion coefficient becomes linear.     

Uniform Asymptotic Expansions of Confluent Hypergeometric Functions

Asymptotic expansions are derived for the confluent hypergeometricfunctions M(a, b, x) and U(a, b, x) for large b. The resultsare uniformly valid with respect to = x/b in a neighbourhoodcontaining = 1; a is a fixed parameter. The expansions arederived from integral representations and contain paraboliccylinder functions and asymptotic series.     

Asymptotic Expansions for Closed Orbits in Homology Classes

In this paper, we study the behaviour of the counting function associated to the closed geodesics lying in a prescribed homology class on a compact negatively curved manifold. Our main result is an asymptotic expansion. We also obtain results in the wider context of periodic orbits of Anosov flows.     

Extended Classical Asymptotic Expansions in the Case of Gaussian Limit Distribution

Let X, X ₁, X ₂,... be a sequence of independent and identically distributed random variables with common distribution function F. Denote by F _n the distribution function of centered and normed sum S _n . Let F belong to the domain of attraction of the standard normal law , that is, lim F _n (x)= (x), as n , uniformly in x . We obtain extended asymptotic expansions for the particular case where the distribution function F has the density p(x) = cx ^––1 ln(x), x > r, where 2, , c > 0, and r > 1. We write the classical asymptotic expansion (in powers of n ^–1/2) and then add new terms of orders n ^–/2 ln n, n ^–/2 ln^-1 n, etc., where 0.     

Asymptotic Expansions for Approximate Eigenvalues of Integral Operators with Nonsmooth Kernels

We consider approximation of eigenvalues of integral operators with Green's function kernels using the Nyström method and the iterated collocation method and obtain asymptotic expansions for approximate eigenvalues. We show that the Richardson extrapolation is applicable to find eigenvalue approximations of higher order and illustrate our results by numerical examples.