Approximate double-transform inversion when time is one of the variables

Abstract

For a continuous-time (integer-time) stochastic process, its distribution at arbitrary time t $(n)$ is often a difficult computation. To use a saddlepoint approximation, its time-indexed moment generating function (MGF) is needed and seldomly is that available. What is often readily available, however, is the Laplace transform (generating function) in time t $(n)$ of this time-indexed MGF which we call a double transform. Such double transforms often take a simple analytic form and we show how they may be inverted to determine the survival function for the process at time t or n. Two general approaches are considered. First, we show that the double-saddlepoint methods initiated by Skovgaard (J. Appl. Probab. 24 (1987) 875–887) may be used by treating the time variable t or n as a random variable with an improper distribution. The second method inverts the double transform in two stages. First, it uses a residue expansion (Butler (J. Appl. Probab. 56 (2019) 307–338; Stoch. Models 39 (2023) 469–501)) to invert it in t or n which is then followed by a single-saddlepoint approximation of the Lugannani-Rice (Adv. in Appl. Probab. 12 (1980) 475–490) type. Applications from renewal theory and renewal reward (cumulative) processes illustrate the remarkable accuracy that results from both of these saddlepoint approximation methods.