Optical solitons in a power law media with fourth order dispersion

In this paper, a closed form optical soliton solution is obtained for the nonlinear Schrödinger’s equation with fourth order dispersion in a power law media. The solitary wave ansatze is used to carry out the integration of this equation. Finally, a numerical simulation is given for the closed form soliton solution.     

Atom-photon interactions with respect to quantum computation: A three-level atom in a two-mode field

In this paper, analysis of a quantum optical system—three-level atom in a quantum electromagnetic field is given. Evolution operators are constructed in closed form. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 44, Quantum Computing, 2007.     

New results on the analytic summation of Adomian series for some classes of differential and integral equations

In this paper, we demonstrate that an infinite number of successive integration by parts can be written in a closed form. This closed form can be used directly to prove that the analytic summation of Adomian series becomes identical to the closed form solution for some classes of differential and integral equations.     

Gaussian Lenses Initiate Optical Vortices in LaserBeams During Self-action in Kerr-like Medium

It is shown that Gaussian lenses initiate closed line of dislocations in coherent beamswith initially smooth wavefront. The stigmatic lenses originate the circle line of edge dislocation,situated on the beam cross-section. In the case of astigmatic Gaussian lens this closed curve getsout from the plane and a quadruple of alternate single-charged optical vortices is born andannihilates in the beam cross-section. The distribution of topological charge signs in the quadruplealong z axis is defined by focusing or defocusing type of Gaussian lens. The model ofGaussian lenses explains self-action of fundamental mode laser beams transmitted by Kerr-likemedia. Two methods were used for study of optical singularities appearance: the geometricaloptics analysis for rapid qualitative location and numerical modeling of Kirchhoff-Fresnel integralfor exact solution.     

Analytical solution for a class of learning by doing models with multiplicative uncertainty

We find the closed form optimal solution for a class of learning by doing models, where multiplicative uncertainty is introduced in a piecewise linear cost reduction function. Previous literature does not find the closed form optimal solution for these models. We consider a monopolist, facing a linear demand function. The optimal policy for the resulting problem is shown to be piecewise linear and continuous. The optimal output increases with unit cost for certain values of the latter. Numerical examples are provided.     

Asymptotic Correction of More Sturm–Liouville Eigenvalue Estimates

The asymptotic correction technique of Paine, de Hoog and Anderssen can dramatically improve the accuracy of finite difference or finite element eigenvalues at negligible extra cost if closed form expressions are available for the errors in a simpler related problem. This paper gives closed form expressions for the errors in the eigenvalues of certain Sturm–Liouville problems obtained by various methods, thereby increasing the range of problems for which asymptotic correction can achieve maximum efficiency. It also investigates implementation of the method for more general problems.     

Exact Closed Form Solution for Two Coupled Inhomogeneous First Order Difference Equations with Variable Coefficients

We present an exact closed form solution for two coupled, homogeneous as well as inhomogeneous, first order difference equations with variable coefficients. The solution is obtained by using the graph theoretic, discrete path formalism. The central parameters in the solution are the crossing index and the crossing number. The transition from an enumerative graph theoretic solution to a closed form combinatorial solution is made possible by an isomorphism in-between paths on the signal flow graph, and n -tuplets of binary numbers.     

Some closed form robust moment‐based estimators for the MEM(1,1)

In this paper, we extend the closed form moment estimator (ordinary MCFE) for the autoregressive conditional duration model given by Lu et al (2016) and propose some closed form robust moment‐based estimators for the multiplicative error model to deal with the additive and innovational outliers. The robustification of the closed form estimator is done by replacing the sample mean and sample autocorrelation with some robust estimators. These estimators are more robust than the quasi‐maximum likelihood estimator (QMLE) often used to estimate this model, and they are easy to implement and do not require the use of any numerical optimization procedure and the choice of initial value. The performance of our proposal in estimating the parameters and forecasting conditional mean μ_t of the MEM(1,1) process is compared with the proposals existing in the literature via Monte Carlo experiments, and the results of these experiments show that our proposal outperforms the ordinary MCFE, QMLE, and least absolute deviation estimator in the presence of outliers in general. Finally, we fit the price durations of IBM stock with the robust closed form estimators and the benchmarks and analyze their performances in estimating model parameters and forecasting the irregularly spaced intraday Value at Risk.     

Synthesis of linear systems with desired equivalent form

A method for feedback synthesis of linear control systems with desired linearly equivalent form of the closed loop system matrix is proposed. The method is based on the serial canonical form of linear multivariable systems which is an alternative to the Luenberger canonical form. A stable computational algorithm for finding the serial canonical form using orthogonal similarity transformations is described. The algorithm for synthesis involves a simple recurrent procedure and gives the possibility to achieve any attainable equivalent form of the closed loop system. The results obtained are extended to the synthesis of reduced order state observers.     

On an analog of periodic boundary value problems for the Poisson equation in the disk

We consider a periodic and an antiperiodic problem for the Poisson equation in the unit disk and prove their well-posedness. The possibility of separation of variables is justified. We construct the Green functions of these problems in closed form and obtain an integral representation of the solution. The problems are self-adjoint, and we construct all eigenvalues and eigenfunctions in closed form.     

Continuation and stability analysis for Bloch waves of the Gross-Pitaevskii equation

We describe a two-parameter continuation algorithm for computing Bloch waves of Bose-Einstein condensates (BEC) in optical lattices which is governed by the Gross-Pitaevskii equation (GPE). The Fourier collocation method and fourth-order Adini’s elements with penalty are used to discretize the GPE. We propose two different approaches so that the two-parameter continuation algorithm can be modified to compute closed tubes at the four corners of the Bloch band. We also study linear stability analysis for the GPE. We show that all the discrete steady-state solutions are numerically neutrally stable. Numerical results show that the four edges of the Bloch waves are surrounded by closed loops if the coefficient of the cubic nonlinear term is greater than that of the periodic potential. Moreover, closed tubes at the four corners of the Bloch band are obtained. The numerical results display superfluidity of BEC.     

Asymptotic Solutions for Australian Options with Low Volatility

Abstract

In this paper we derive asymptotic expansions for Australian options in the case of low volatility using the method of matched asymptotics. The expansion is performed on a volatility scaled parameter. We obtain a solution that is of up to the third order. In case that there is no drift in the underlying, the solution provided is in closed form, for a non-zero drift, all except one of the components of the solutions are in closed form. Additionally, we show that in some non-zero drift cases, the solution can be further simplified and in fact written in closed form as well. Numerical experiments show that the asymptotic solutions derived here are quite accurate for low volatility.     

Bonds and Options in Exponentially Affine Bond Models

Abstract

In this article we apply the Flesaker–Hughston approach to invert the yield curve and to price various options by letting the randomness in the economy be driven by a process closely related to the short rate, called the abstract short rate. This process is a pure deterministic translation of the short rate itself, and we use the deterministic shift to calibrate the models to the initial yield curve. We show that we can solve for the shift needed in closed form by transforming the problem to a new probability measure. Furthermore, when the abstract short rate follows a Cox–Ingersoll–Ross (CIR) process we compute bond option and swaption prices in closed form. We also propose a short-rate specification under the risk-neutral measure that allows the yield curve to be inverted and is consistent with the CIR dynamics for the abstract short rate, thus giving rise to closed form bond option and swaption prices.     

Magnetic Energies and Feynman–Kac–Itô Formulas for Symmetric Markov Processes

Given a (conservative) symmetric Markov process on a metric space we consider related bilinear forms that generalize the energy form for a particle in an electromagnetic field. We obtain one bilinear form by semigroup approximation and another, closed one, by using a Feynman–Kac–Itô formula. If the given process is Feller, its energy measures have densities and its jump measure has a kernel, then the two forms agree on a core and the second is a closed extension of the first. In this case we provide the explicit form of the associated Hamiltonian.     

Analysis of transient queues with semidefinite optimization

We derive upper bounds on the tail distribution of the transient waiting time in the GI/GI/1 queue, given a truncated sequence of the moments of the service time and that of the interarrival time. Our upper bound is given as the objective value of the optimal solution to a semidefinite program (SDP) and can be calculated numerically by solving the SDP. We also derive the upper bounds in closed form for the case when only the first two moments of the service time and those of the interarrival time are given. The upper bounds in closed form are constructed by formulating the dual problem associated with the SDP. Specifically, we obtain the objective value of a feasible solution of the dual problem in closed from, which turns out to be the upper bound that we derive. In addition, we study bounds on the maximum waiting time in the first busy period.     

Optimal Exercise Strategies for Operational Risk Insurance via Multiple Stopping Times

In this paper we demonstrate how to develop analytic closed form solutions to optimal multiple stopping time problems arising in the setting in which the value function acts on a compound process that is modified by the actions taken at the stopping times. This class of problem is particularly relevant in insurance and risk management settings and we demonstrate this on an important application domain based on insurance strategies in Operational Risk management for financial institutions. In this area of risk management the most prevalent class of loss process models is the Loss Distribution Approach (LDA) framework which involves modelling annual losses via a compound process. Given an LDA model framework, we consider Operational Risk insurance products that mitigate the risk for such loss processes and may reduce capital requirements. In particular, we consider insurance products that grant the policy holder the right to insure k of its annual Operational losses in a horizon of T years. We consider two insurance product structures and two general model settings, the first are families of relevant LDA loss models that we can obtain closed form optimal stopping rules for under each generic insurance mitigation structure and then secondly classes of LDA models for which we can develop closed form approximations of the optimal stopping rules. In particular, for losses following a compound Poisson process with jump size given by an Inverse-Gaussian distribution and two generic types of insurance mitigation, we are able to derive analytic expressions for the loss process modified by the insurance application, as well as closed form solutions for the optimal multiple stopping rules in discrete time (annually). When the combination of insurance mitigation and jump size distribution does not lead to tractable stopping rules we develop a principled class of closed form approximations to the optimal decision rule. These approximations are developed based on a class of orthogonal Askey polynomial series basis expansion representations of the annual loss compound process distribution and functions of this annual loss.     

Some cases of the solvability in a closed form of singular integral equations and two-dimensional characteristic systems

We consider the characteristic system of singular integral equations for two unknown functions. We obtain certain correlations connecting coefficients of the system which allow one to solve the latter in a closed form. Based on the obtained results, we consider cases of the solvability in a closed form for one class of singular integral equations.     

Dual control Monte-Carlo method for tight bounds of value function under Heston stochastic volatility model

The aim of this paper is to study the fast computation of the lower and upper bounds on the value function for utility maximization under the Heston stochastic volatility model with general utility functions. It is well known there is a closed form solution to the HJB equation for power utility due to its homothetic property. It is not possible to get closed form solution for general utilities and there is little literature on the numerical scheme to solve the HJB equation for the Heston model. In this paper we propose an efficient dual control Monte-Carlo method for computing tight lower and upper bounds of the value function. We identify a particular form of the dual control which leads to the closed form upper bound for a class of utility functions, including power, non-HARA and Yaari utilities. Finally, we perform some numerical tests to see the efficiency, accuracy, and robustness of the method. The numerical results support strongly our proposed scheme.     

Integrals,infinite series and closed forms

A number of examples of infinite series are given whose sum can be expressed in closed form, sometimes as an integral. This leads naturally to the determination of a closed form for the sum of a less well-known series in terms of an interesting definite integral. Other examples are included, as well as suggestions for further work.     

The codimension of singular matrix pairs

The singular pairs of n × n matrices [those satisfying det(A? λB)  0] form a closed set of codimension n + 1 inside the space of all matrix pairs. The same holds for singular symmetric pairs. For Hermitian pairs, the singular ones form a closed set of codimension n+ 1 orn + 2 according as n is odd or even. The irreducible components of these closed sets are determined by various basic singular summands.