scholarly journals Approximating the Laplace transform of the sum of dependent lognormals

2016 ◽  
Vol 48 (A) ◽  
pp. 203-215 ◽  
Author(s):  
Patrick J. Laub ◽  
Søren Asmussen ◽  
Jens L. Jensen ◽  
Leonardo Rojas-Nandayapa
Keyword(s):  
Monte Carlo ◽  
Laplace Transform ◽  
Taylor Expansion ◽  
Multivariate Normal ◽  
Numerical Examples ◽  
Quasi Monte Carlo ◽  
Error Factor ◽  
The Laplace Transform ◽  
Laplace Transform Inversion ◽  
Mean Vector

AbstractLet (X1,...,Xn) be multivariate normal, with mean vector 𝛍 and covariance matrix 𝚺, and letSn=eX1+⋯+eXn. The Laplace transform ℒ(θ)=𝔼e-θSn∝∫exp{-hθ(𝒙)}d𝒙 is represented as ℒ̃(θ)I(θ), where ℒ̃(θ) is given in closed form andI(θ) is the error factor (≈1). We obtain ℒ̃(θ) by replacinghθ(𝒙) with a second-order Taylor expansion around its minimiser 𝒙*. An algorithm for calculating the asymptotic expansion of 𝒙*is presented, and it is shown thatI(θ)→ 1 as θ→∞. A variety of numerical methods for evaluatingI(θ) is discussed, including Monte Carlo with importance sampling and quasi-Monte Carlo. Numerical examples (including Laplace-transform inversion for the density ofSn) are also given.

The solution of theoretical seismic problems by best rational function approximations for Laplace transform inversion

1975 ◽  
Vol 65 (4) ◽  
pp. 927-935
Author(s):  
I. M. Longman ◽  
T. Beer
Keyword(s):  
Laplace Transform ◽  
Rational Function ◽  
High Accuracy ◽  
Time Variable ◽  
Inversion Method ◽  
The Laplace Transform ◽  
Previously Treated ◽  
The Mean ◽  
Function Approximations ◽  
Laplace Transform Inversion

Abstract In a recent paper, the first author has developed a method of computation of “best” rational function approximations ḡn(p) to a given function f̄(p) of the Laplace transform operator p. These approximations are best in the sense that analytic inversion of ḡn(p) gives a function gn(t) of the time variable t, which approximates the (generally unknown) inverse f(t) of f̄(p in a minimum least-squares manner. Only f̄(p) but not f(t) is required to be known in order to carry out this process. n is the “order” of the approximation, and it can be shown that as n tends to infinity gn(t) tends to f(t) in the mean. Under suitable conditions on f(t) the convergence is extremely rapid, and quite low values of n (four or five, say) are sufficient to give high accuracy for all t ≧ 0. For seismological applications, we use geometrical optics to subtract out of f(t) its discontinuities, and bring it to a form in which the above inversion method is very rapidly convergent. This modification is of course carried out (suitably transformed) on f̄(p), and the discontinuities are restored to f(t) after the inversion. An application is given to an example previously treated by the first author by a different method, and it is a certain vindication of the present method that an error in the previously given solution is brought to light. The paper also presents a new analytical method for handling the Bessel function integrals that occur in theoretical seismic problems related to layered media.

Optimal importance sampling for the Laplace transform of exponential Brownian functionals

2016 ◽  
Vol 53 (2) ◽  
pp. 531-542 ◽  
Author(s):  
Je Guk Kim
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Performance Analysis ◽  
Laplace Transform ◽  
Importance Sampling ◽  
Pricing Model ◽  
Asymptotically Optimal ◽  
Closed Form Solutions ◽  
The Laplace Transform ◽  
Exponential Brownian Functionals

Abstract We present an asymptotically optimal importance sampling for Monte Carlo simulation of the Laplace transform of exponential Brownian functionals which plays a prominent role in many disciplines. To this end we utilize the theory of large deviations to reduce finding an asymptotically optimal importance sampling measure to solving a calculus of variations problem. Closed-form solutions are obtained. In addition we also present a path to the test of regularity of optimal drift which is an issue in implementing the proposed method. The performance analysis of the method is provided through the Dothan bond pricing model.

On the recovery and resolution of exponential relaxation rates from experimental data II. The optimum choice of experimental sampling points for Laplace transform inversion

1984 ◽  
Vol 393 (1804) ◽  
pp. 51-65 ◽  
Keyword(s):  
Experimental Data ◽  
Laplace Transform ◽  
Upper And Lower Bounds ◽  
Relaxation Rates ◽  
Scientific Experiments ◽  
The Laplace Transform ◽  
Data Points ◽  
Exponential Relaxation ◽  
Ill Conditioning ◽  
Laplace Transform Inversion

In a previous paper (Bertero, Boccacci & Pike, Proc. R. Soc. Lond . A 383, 15 (1982)) we investigated the benefits of a priori knowledge of finite support of the solution in the inversion of the Laplace transform and determined the number of exponential components that could be recovered from given data in the presence of quantified amounts of noise. The support of the data was assumed to be unrestricted. In the present paper these calculations are extended to cover the case of a necessarily finite number of experimental data points. We consider two cases: uniform distribution and geometric distribution of data points. In both cases the condition number of singular value inversions is minimized with respect to the placing of the data points for fixed numbers of data samples. The results allow optimum placing of the experimental data points as a function of the ratio, γ , of given upper and lower bounds of the support of the solution. The number of exponentials recoverable is less than in the unrestricted case but for γ ≼ 8 we find that with 32 linearly spaced data points, optimally placed, this reduction is rather small. Geometrical sampling of experimental data reduces further the number of sample points required for inversion. For γ ≼ 8 we show that with 5 geometrically spaced data points, points, optimally placed, the ill-conditioning in the restoration of 2 to 4 exponential components is smaller than with 32 linearly spaced data samples. This has important implications for the design of a number of scientific experiments and instruments.

scholarly journals Classroom note: Fourier method for Laplace transform inversion

2001 ◽  
Vol 5 (3) ◽  
pp. 193-200 ◽  
Author(s):  
M. Iqbal
Keyword(s):  
Laplace Transform ◽  
Fourier Method ◽  
Test Functions ◽  
The Laplace Transform ◽  
Laplace Transform Inversion

A method is described for inverting the Laplace transform. The performance of the Fourier method is illustrated by the inversion of the test functions available in the literature. Results are shown in the tables.

scholarly journals Descriptor fractional linear systems with regular pencils

2013 ◽  
Vol 23 (2) ◽  
pp. 309-315 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Laplace Transform ◽  
Linear Systems ◽  
Discrete Time ◽  
Continuous Time ◽  
The State ◽  
Transition Matrices ◽  
State Equations ◽  
Numerical Examples ◽  
The Laplace Transform ◽  
Time Linear

Methods for finding solutions of the state equations of descriptor fractional discrete-time and continuous-time linear systems with regular pencils are proposed. The derivation of the solution formulas is based on the application of the Z transform, the Laplace transform and the convolution theorems. Procedures for computation of the transition matrices are proposed. The efficiency of the proposed methods is demonstrated on simple numerical examples.

scholarly journals Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales

2008 ◽  
Vol 45 (02) ◽  
pp. 531-541 ◽  
Author(s):  
A. G. Rossberg
Keyword(s):  
Laplace Transform ◽  
High Precision ◽  
Computational Efficiency ◽  
Generating Functions ◽  
Heavy Tails ◽  
Probability Distributions ◽  
Random Variables ◽  
Numerical Examples ◽  
Pareto Distributions ◽  
The Laplace Transform

It is shown that, when expressing arguments in terms of their logarithms, the Laplace transform of a function is related to the antiderivative of this function by a simple convolution. This allows efficient numerical computations of moment generating functions of positive random variables and their inversion. The application of the method is straightforward, apart from the necessity to implement it using high-precision arithmetics. In numerical examples the approach is demonstrated to be particularly useful for distributions with heavy tails, such as lognormal, Weibull, or Pareto distributions, which are otherwise difficult to handle. The computational efficiency compared to other methods is demonstrated for an M/G/1 queueing problem.

A Technique for Increasing the Accuracy of the Numerical Inversion of the Laplace Transform With Applications

1973 ◽  
Vol 40 (4) ◽  
pp. 1110-1112 ◽  
Author(s):  
B. S. Berger ◽  
S. Duangudom
Keyword(s):  
Laplace Transform ◽  
Initial Value Problems ◽  
Numerical Inversion ◽  
Initial Value ◽  
Numerical Examples ◽  
Oscillatory Function ◽  
Higher Order Terms ◽  
The Laplace Transform ◽  
Image Function ◽  
Inversion Techniques

A technique is introduced which extends the range of useful approximation of numerical inversion techniques to many cycles of an oscillatory function without requiring either the evaluation of the image function for many values of s or the computation of higher-order terms. The technique consists in reducing a given initial value problem defined over some interval into a sequence of initial value problems defined over a set of subintervals. Several numerical examples demonstrate the utility of the method.

scholarly journals Combination of Laplace transform and residual power series techniques to solve autonomous n-dimensional fractional nonlinear systems

2021 ◽  
Vol 10 (1) ◽  
pp. 282-292
Author(s):  
Marwan Alquran ◽  
Maysa Alsukhour ◽  
Mohammed Ali ◽  
Imad Jaradat
Keyword(s):  
Nonlinear Systems ◽  
Power Series ◽  
Laplace Transform ◽  
Iterative Algorithm ◽  
Autonomous Systems ◽  
Numerical Examples ◽  
The Laplace Transform ◽  
Residual Power Series ◽  
The Impact ◽  
Residual Power

Abstract In this work, a new iterative algorithm is presented to solve autonomous n-dimensional fractional nonlinear systems analytically. The suggested scheme is combination of two methods; the Laplace transform and the residual power series. The methodology of this algorithm is presented in details. For the accuracy and effectiveness purposes, two numerical examples are discussed. Finally, the impact of the fractional order acting on these autonomous systems is investigated using graphs and tables.

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