New quadrature formulas for the numerical inversion of the Laplace transform

New quadrature formulas for the evaluation of the Bromwich integral, arising in the inversion of the Laplace transform are discussed. They are obtained by optimal addition of abscissas to Gaussian quadrature formulas. A table of abscissas and weights is given.     

A NEW INTEGRAL TRANSFORM AND ITS APPLICATIONS

In the present paper,the authors introduce a new integral transform which yields a number of potentially useful(known or new) integral transfoms as its special cases.Many fundamental results about this new integral transform,which are established in this paper,include(for example) existence theorem,Parseval-type relationship and inversion formula.The relationship between the new integral transform with the H-function and the H-transform are characterized by means of some integral identities.The introduced transform is also used to find solution to a certain differential equation.Some illustrative examples are also given.     

An Error Analysis and Convergence of a Quadrature Formula to Invert Laplace Transforms

The use of an expression for the exponential function exp(st)in Bromwich's integral, which incorporates a Padé approximation,yields a well known quadrature formula and its error terms.The error is then analysed for rational, transcendental andnon-rational Laplace transforms respectively and a convergencescheme is presented for rational transforms.     

Numerical solution of modified Black–Scholes equation pricing stock options with discrete dividend

This paper deals with the numerical solution of the modified Black–Scholes equation modelling the valuation of stock options with discrete dividend payments. By using a delta-defining sequence of the involved generalized Dirac delta function and applying the Mellin transform, an integral formula for the solution is obtained. Then, numerical quadrature approximations and illustrative examples are given.     

Lubich convolution quadratures and their application to problems described by space-time BIEs

In the last years several authors have used Lubich convolution quadrature formulas to discretize space-time boundary integral equations representing time dependent problems. These rules have the fundamental property of not using explicitly the expression of the kernel of the integral equation they are applied to, which is instead replaced by that of its Laplace transform, usually given by a simple analytic function. In this paper, a review of these rules, which includes their main properties, several new remarks and some conjectures, will be presented when they are applied to the heat and wave space-time boundary integral equation formulations. The construction and behavior of the corresponding coefficients are analyzed and tested numerically. When the quadrature is defined by a BDF method, a new approach for the representation of its coefficients is presented.     

Special quadrature rules for Laplace transform inversion

Quadrature rules for Laplace transform inversion are studied that are adapted to the inversion of transforms corresponding to slowly varying long processes characteristic of linear viscoelasticity problems. The convergence of special quadrature rules for Laplace transform inversion is proved.     

基于直接数值积分的Laplace逆变换方法的比较研究

为了探讨各种数值积分方法,如梯形公式、Simpson法、Gauss积分方法和振荡函数积分方法等,在数值Laplace逆变换中的应用效果,本文进行了基于各种离散数值积分公式的Laplace逆变换方法的比较研究,涉及到24种方法,针对Davies和Martin的16个考题,给出了数值比较结果,得出了一些新的结论。     

Efficient Computation of First Passage Times in Kou’s Jump-diffusion Model

S. G. Kou and H. Wang [First passage times of a jump diffusion process, Adv. Appl. Probab. 35 (2003) 504–531] give expressions of both the real Laplace transform of the distribution of first passage time and the real Laplace transform of the joint distribution of the first passage time and the running maxima of a jump-diffusion model called Kou model. These authors invert the former Laplace transform by using Gaver-Stehfest algorithm, and for the latter they need a large computing time with an algebra computer system. In the present paper, we give a much simpler expression of the Laplace transform of the joint distribution, and we also show, using Complex Analysis techniques, that both Laplace transforms can be extended to the complex plane. Hence, we can use inversion methods based on the complex inversion formula or Bromwich integral which are very efficent. The improvement in the computing times and accuracy is remarkable.     

Contour integral method for European options with jumps

We develop an efficient method for pricing European options with jump on a single asset. Our approach is based on the combination of two powerful numerical methods, the spectral domain decomposition method and the Laplace transform method. The domain decomposition method divides the original domain into sub-domains where the solution is approximated by using piecewise high order rational interpolants on a Chebyshev grid points. This set of points are suitable for the approximation of the convolution integral using Gauss–Legendre quadrature method. The resulting discrete problem is solved by the numerical inverse Laplace transform using the Bromwich contour integral approach. Through rigorous error analysis, we determine the optimal contour on which the integral is evaluated. The numerical results obtained are compared with those obtained from conventional methods such as Crank–Nicholson and finite difference. The new approach exhibits spectrally accurate results for the evaluation of options and associated Greeks. The proposed method is very efficient in the sense that we can achieve higher order accuracy on a coarse grid, whereas traditional methods would required significantly more time-steps and large number of grid points.     

The Fourier-series method for inverting transforms of probability distributions

This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this purpose, we also describe two methods for inverting Laplace transforms based on the Post-Widder inversion formula. The overall procedure is illustrated by several queueing examples.     

A discrete Hartley transform based on Simpson's rule

The Hartley transform is an integral transformation that maps a real valued function into a real valued frequency function via the Hartley kernel, thereby avoiding complex arithmetic as opposed to the Fourier transform. Approximation of the Hartley integral by the trapezoidal quadrature results in the discrete Hartley transform, which has proven a contender to the discrete Fourier transform because of its involutory nature. In this paper, a discrete transform is proposed as a real transform with a convolution property and is an alternative to the discrete Hartley transform. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical method for the solution of integral equations in a problem with directional derivative for the Laplace equation outside open curves

By using a simple layer potential and an angular potential, one can reduce the problem with a directional derivative for the Laplace equation outside several open curves on the plane to a uniquely solvable system of integral equations that consists of an integral equation of the second kind and additional integral conditions. The kernel in the integral equation of the second kind contains singularities and can be represented as a Cauchy singular integral. We suggest a numerical method for solving a system of integral equations. Quadrature formulas for the logarithmic and angular potentials are represented. The quadrature formula for the logarithmic potential preserves the property of its continuity across the boundary (open curves).     

Numerical Treatment of Homogeneous Semi-Markov Processes in Transient Case–a Straightforward Approach

This paper presents the numerical solution of the process evolution equation of a homogeneous semi-Markov process (HSMP) with a general quadrature method. Furthermore, results that justify this approach proving that the numerical solution tends to the evolution equation of the continuous time HSMP are given. The results obtained generalize classical results on integral equation numerical solutions applying them to particular kinds of integral equation systems. A method for obtaining the discrete time HSMP is shown by applying a very particular quadrature formula for the discretization. Following that, the problem of obtaining the continuous time HSMP from the discrete one is considered. In addition, the discrete time HSMP in matrix form is presented and the fact that the solution of the evolution equation of this process always exists is proved. Afterwards, an algorithm for solving the discrete time HSMP is given. Finally, a simple application of the HSMP is given for a real data social security example.     

On the construction of quadrature rules for Laplace transform inversion

For Laplace transform inversion, a method for constructing quadrature rules of the highest degree of accuracy based on an asymptotic distribution of roots of special orthogonal polynomials on the complex plane is proposed.     

A note on the discrete Cauchy‐Kovalevskaya extension

In this paper, we exploit the umbral calculus framework to reformulate the so‐called discrete Cauchy‐Kovalevskaya extension in the scope of hypercomplex variables. The key idea is to consider not only formal power series representation for the underlying solution, but also integral representations for the Chebyshev polynomials of first and second kind by means of its Cauchy principal values. It turns out that the resulting integral representation associated to our toy problem is a space‐time Fourier type inversion formula. Moreover, with the aid of some Laplace transform identities involving the generalized Mittag‐Leffler function, we are able to establish a link with a Cauchy problem of differential‐difference type.     

On the functional properties of the confluent hypergeometric zeta-function

A zeta-function associated with Kummer’s confluent hypergeometric function is introduced as a classical Dirichlet series. An integral representation, a transformation formula, and relation formulas between contiguous functions and one generalization of Ramanujan’s formula are given. The inverse Laplace transform of confluent hypergeometric functions is essentially used to derive the integral representation.     

A note on the inversion of Laplace transforms

A direct method, using a Mellin transform technique, is presented to derive the solution of a special class of first kind integral equations over the positive real axis, and as a particular case, an inversion formula is deduced for the Laplace transform F(p) of a function f(x) (x>0), when F(p) is known only for p>0.     

On the Accuracy of Quadrature Laplace Transform Inversion and the Solution of State Space Equations

A quadrature formula is shown to be an approximation of thepower-series method of inverting Laplace transforms. This togetherwith the properties of the constants derived from a power-seriesexpansion of the Pad? approximation to exp (s) yield an importantupper limit on t which is quite sharp in determining the breakdownpoint up to and after which the approximation is accurate andinaccurate respectively. The solution of state space equationsusing the quadrature inversion formula is also discussed.     

Determination of nonstationary temperature fields and stresses in piecewise homogeneous circular plates on the basis of a numerical-analytic Laplace inversion formula

We investigate nonstationary temperature fields and stresses generated by them in piecewise homogeneous annular plates. An algorithm for the solution of the problem is based on the direct calculation of the Laplace transform and a modified Prudnikov formula for its inversion.     

Asymptotic Estimates of the Distribution of Brownian Hitting Time of a Disc

The distribution of the first hitting time of a disc for the standard two-dimensional Brownian motion is computed. By investigating the inversion integral of its Laplace transform we give fairly detailed asymptotic estimates of its density valid uniformly with respect to the point from which the Brownian motion starts.