Evaluating Scale Functions of Spectrally Negative Lévy Processes

Abstract

In this paper we present a robust numerical method to compute the scale function W^(q)(x) of a general spectrally negative Lévy process (X, P). The method is based on the Esscher transform of measure P^ν under which X is taken and the scale function is determined. This change of measure makes it possible for the scale function to be bounded and, hence, makes numerical computation easy, fast, and stable. Working under the new measure P^ν and using the method of Abate and Whitt (1992) and Choudhury, Lucantoni, and Whitt (1994), we give a fast stable numerical algorithm for the computation of W^(q)(x).