Numerical Inversion of Laplace Transforms of Probability Distributions

Abstract

We present a simple algorithm for numerically inverting Laplace transforms. The algorithm is designed especially for probability cumulative distribution functions, but it applies to other functions as well. Since it does not seem possible to provide effective methods with simple general error bounds, we simultaneously use two different methods to confirm the accuracy. Both methods are variants of the Fourier-series method. The first, building on Dubner and Abate (Dubner, H., J. Abate. 1968. Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. JACM 15 115–123.) and Simon, Stroot, and Weiss (Simon, R. M., M. T. Stroot, G. H. Weiss. 1972. Numerical inversion of Laplace transforms with application to percentage labeled experiments. Comput. Biomed. Res. 6 596–607.), uses the Bromwich integral, the Poisson summation formula and Euler summation; the second, building on Jagerman (Jagerman, D. L. 1978. An inversion technique for the Laplace transform with applications. Bell System Tech. J. 57 669–710 and Jagerman, D. L. 1982. An inversion technique for the Laplace transform. Bell System Tech. J. 61 1995–2002.), uses the Post-Widder formula, the Poisson summation formula, and the Stehfest (Stehfest, H. 1970. Algorithm 368. Numerical inversion of Laplace transforms. Comm. ACM 13 479–490 (erratum: 13 624).) enhancement. The resulting program is short and the computational experience is encouraging. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.