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

This article provides the saddlepoint approximations to the density and to 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 an alternative method, based on the Padé approximation, that likewise possesses the advantage of being expressible in closed-form. Numerical 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 conclusion is that, although no universally optimal method for inverting Laplace transforms exists, the saddlepoint method, when applicable, that is, when a light tail condition on the unknown distribution is fulfilled, is among the preferred techniques of Laplace transform inversion. 2000 MSC: 41A60, 60G40, 60G51