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...

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Many methods for numerically inverting transforms require values of the transform at complex arguments. However, in some applications, the transforms are only characterized implicitly via functional equations. This is illustrated by the busy-period distribution in the M/G/1 queue. In this paper we provide conditions for iterative methods to converge for complex arguments. Moreover, we show that stochastic monotonicity properties can provide useful bounds.

We introduce and investigate a framework for constructing algorithms to numerically invert Laplace transforms. Given a Laplace transform ˆ f of a complex-valued function of a nonneg-ative real-variable, f, the function f is approximated by a finite linear combination of the transform values; i.e., we use the inversion formula f(t) ≈ fn(t) ≡ 1 t n� ωk ˆ f k=0 αk

Resource allocation is a fundamental but challenging problem due to the complexity of cluster computing systems. In enterprise service computing, resource allocation is often associated with a service level agreement (SLA) which is a set of quality of services and a price agreed between a customer and a service provider. The SLA plays an important role in an e-business application. A service provider uses a set of computer resources to support e-business applications subject to an SLA. In this paper, we present an approach for computer resource allocation in such an environment that minimizes the total cost of computer resources used by a service provider for an e-business application while satisfying the quality of service (QoS) defined in an SLA. These QoS metrics include percentile response time, cluster utilization, packet loss rate and cluster availability. Simulation results show the applicability of the approach and validate its accuracy. I.

Installment options are weakly path-dependent contingent claims in which the premium is paid discretely or continuously in installments, instead of paying a lump sum at the time of purchase. This paper deals with valuing American continuousinstallment options written on dividend-paying assets. The setup is a standard Black-Scholes-Merton framework where the price of the underlying asset evolves according to a geometric Brownian motion. The valuation of installment options can be formulated as an optimal stopping problem, due to the flexibility of continuing or stopping to pay installments as well as the chance of early exercise. Analyzing cash flow generated by the optimal stop, we can characterize asymptotic behaviors of the stopping and early exercise boundaries close to expiry. Combining the PDE and Laplace transform approaches, we obtain explicit Laplace transforms of the initial premium as well as its Greeks, which include the transformed stopping and early exercise boundaries. Abelian theorems of Laplace transforms enable us to obtain a concise result for the perpetual case. We show that numerical inversion of these Laplace transforms works well for computing both the option value and the boundaries.

The numerical inversion of Laplace transform arises in many applications of science and engineering whenever ordinary and partial differential equations or integral equations are solved. The increasing number of available numerical methods and computer codes has generated a need for well-documented sets of test problems. Using such sets, algorithm developers can evaluate the relative merits and drawbacks of their suggested new methods, and end-users can make judgments on the applicability of an individual method for a specific problem. Many areas in science and engineering, lead to problems that share three important properties: i) the image function can be evaluated for real arguments, but not necessarily for complex ones; ii) the original is known to be infinitely differentiable for times t> 0, iii) the values of the image function can be obtained with any prescribed accuracy. The published test sets do not properly cover these applications, as many included problems are beyond of the specific class, while the remaining ones fail to address some of the potential difficulties arising in practice. The goal of this paper is to establish a common ground for problem classification, to list the requirements for the above class of problems, and to provide a carefully selected test set by addressing the

This is a supplement to the main paper, presenting additional experimental results not included in the main paper due to lack of space. The overall study investigates ways to create algorithms to numerically invert Laplace transforms within a unified framework proposed by Abate and Whitt (2006). That framework approximates the desired function value by a finite linear combination of transform values, depending on parameters called weights and nodes, which are initially left unspecified. Alternative parameter sets, and thus algorithms, are generated and evaluated here by considering power test functions. The resulting power algorithms are shown to be effective, with the parameter choice being tunable to the trans-form being inverted. The powers can be advantageously chosen from series expansions of the transform. Real weights for a real-variable power algorithm are found for specified real powers and positive real nodes by solving a system of linear equations involving a generalized Vandermonde matrix, using Mathematica. Experiments show that the power algorithms are robust in the nodes; it suffices to use the first n positive integers. The power test functions also provide a useful way to evaluate the performance of other algorithms. History: draft aiming for JoC. Last modified on December 9, 2006.

In most cases, the transition density function of an Itô stochastic differential equation is not available in closed-form. Using Feynman-Kac integration, we construct an exact recursion scheme for the Laplace transform of the transition density. This allows a very accurate and nearly analytical treatment of a wide range of valuation and econometric problems. Generalizations of our technique to functionals of Lévy processes are also briefly discussed.