Saddlepoint Approximation to the Distribution of the Absorption Time of the Diffusion under Resetting and Reflection

Abstract This article provides the saddlepoint approximations to the density and the upper tail probabilities of the first passage time to the null level of the drifted Wiener diffusion, bounded by an upper hard-wall barrier and subject to resetting at the initial value occurring at Poisson times. The proposed saddlepoint approximation is compared with Monte Carlo and with an alternative method of Laplace transform inversion based on the Padé approximation, which likewise possesses the advantage of being expressible in closed-form. These comparisons illustrate the high accuracy of the saddlepoint approximation, in particular in terms of relative errors and for small upper tail probabilities. This high level of accuracy is achieved with a computational cost that is substantially lower than that of its competing methods. The overall conclusion is that, although no universally optimal procedure for the inversion of Laplace transforms can be identified, the saddlepoint method, when applicable (i.e., when a steepness condition on the Laplace transform is satisfied), is one of the most advantageous techniques for Laplace transform inversion.