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...

Abstract not found

The modelling of credit events is in effect the modelling of the times to default of various names. The distribution of individual times to default can be calibrated from CDS quotes, but for more complicated instruments, such as CDOs, the joint law is needed. Industry practice is to model this correlation using a copula or base correlation approach, both of which suffer significant deficiencies. We present a new approach to default correlation modelling, where defaults of different names are driven by a common continuous-time Markov process. Individual default probabilities and default correlations can be calculated in closed form. As illustrations, CDO tranches with name-dependent random losses are computed using Laplace transform techniques. The model is calibrated to standard tranche spreads with encouraging results.

Abstract not found

We present a general framework to price contingent claims whose payoffs involve equity, credit and interest rate components. The common cross-market dynamics are modeled via a Markov-chain ξ. The model is dynamically consistent and allows for a high degree of flexibility. Prices of various vanilla and more complex derivative products can be derived analytically or resorting to integral transform techniques. 1

doi:10.3906/elk-0905-2 State variable distributed-parameter representation of transmission line for transient simulations

In this paper we study the transient behavior of the MGE L/MGEM/1 queueing system, where MGE is the class of mixed generalized Erlang distributions which can approximate an arbitrary distribution. We use the method of stages combined with the separation of variables and root finding techniques together with linear and tensor algebra. We find simple closed form expressions for the Laplace transforms of the queue length distribution and the waiting time distribution under FCFS when the system is initially empty and the busy period distribution. We report computational results by inverting these expressions numerically in the time domain. Because of the simplicity of the expressions derived our algorithm is very fast and robust.

Three computational methods exist for time-domain optics: time domain, frequency domain, and hybrid time–frequency domain. Because of temporal dispersion in optical materials, problems unique to ultrafast optics require well-defined and highly accurate approximation methods.