Fast Laplace transform methods for the PDE system of Parisian and Parasian option pricing

This paper develops a fast Laplace transform method for solving the complex PDE system arising from Parisian and Parasian option pricing.The value functions of the options are governed by a system of partial differential equations(PDEs)of two and three dimensions.Applying the Laplace transform to the PDEs with respect to the calendar time to maturity leads to a coupled system consisting of an ordinary differential equation(ODE)and a 2-dimensional partial differential equation(2d-PDE).The solution to this ODE is found analytically on a specific parabola contour that is used in the fast Laplace inversion,whereas the solution to the 2d-PDE is approximated by solving 1-dimensional integro-differential equations.The Laplace inversion is realized by the fast contour integral methods.Numerical results confirm that the Laplace transform methods have the exponential convergence rates and are more efficient than the implicit finite difference methods,Monte Carlo methods and moving window methods.

A Regularization Method for Approximating the Inverse Laplace Transform

A new method for approximating the inerse Laplace transform is presented. We first change our Laplace transform equation into a convolution type integral equation, where Tikhonov regularization techniques and the Fourier transformation are easily applied. We finally obtain a regularized approximation to the inverse Laplace transform as finite sum

New homotopy analysis transform method for solving the discontinued problems arising in nanotechnology

We present a new reliable analytical study for solving the discontinued problems arising in nanotechnology. Such problems are presented as nonlinear differential-difference equations. The proposed method is based on the Laplace trans- form with the homotopy analysis method （HAM）. This method is a powerful tool for solving a large amount of problems. This technique provides a series of functions which may converge to the exact solution of the problem. A good agreement between the obtained solution and some well-known results is obtained.

Analysis for pressure transient of coalbed methane reservoir based on Laplace transform finite difference method

Based on fractal geometry,fractal medium of coalbed methane mathematical model is established by Langmuir isotherm adsorption formula,Fick's diffusion law,Laplace transform formula,considering the well bore storage effect and skin effect.The Laplace transform finite difference method is used to solve the mathematical model.With Stehfest numerical inversion,the distribution of dimensionless well bore flowing pressure and its derivative was obtained in real space.According to compare with the results from the analytical method,the result from Laplace transform finite difference method turns out to be accurate.The influence factors are analyzed,including fractal dimension,fractal index,skin factor,well bore storage coefficient,energy storage ratio,interporosity flow coefficient and the adsorption factor.The calculating error of Laplace transform difference method is small.Laplace transform difference method has advantages in well-test application since any moment simulation does not rely on other moment results and space grid.

The bound state solution for the Morse potential with a localized mass profile

We investigate an analytical solution for the Schr o¨dinger equation with a position-dependent mass distribution, with the Morse potential via Laplace transformations. We considered a mass function localized around the equilibrium position.The mass distribution depends on the energy spectrum of the state and the intrinsic parameters of the Morse potential. An exact bound state solution is obtained in the presence of this mass distribution.

A Numerical Algorithm for Arbitrary Real-Order Hankel Transform

The Hankel transform is widely used to solve various engineering and physics problems,such as the representation of electromagnetic field components in the medium,the representation of dynamic stress intensity factors,vibration of axisymmetric infinite membrane and displacement intensity factors which all involve this type of integration.However,traditional numerical integration algorithms cannot be used due to the high oscillation characteristics of the Bessel function,so it is particularly important to propose a high precision and efficient numerical algorithm for calculating the integral of high oscillation.In this paper,the improved Gaver-Stehfest(G-S)inverse Laplace transform method for arbitrary real-order Bessel function integration is presented by using the asymptotic characteristics of the Bessel function and the accumulation of integration,and the optimized G-S coefficients are given.The effectiveness of the algorithm is verified by numerical examples.Compared with the linear transformation accelerated convergence algorithm,it shows that the G-S inverse Laplace transform method is suitable for arbitrary real order Hankel transform,and the time consumption is relatively stable and short,which provides a reliable calculation method for the study of electromagnetic mechanics,wave propagation,and fracture dynamics.