On approximations for the distribution of the time of first level crossing

Abstract: This paper is an overview of the classical level crossing problem which is studied extensively in the literature and is fundamental in many branches of applied probability. We discuss a number of approximations with an emphasis on their performance, methods of justification and technical conditions which are required in these methods, including a new approximation called "inverse Gaussian". It is derived by a new method, is fruitful for solving related problems, and is valid under mild regularity conditions. We emphasize its novelty and boons.