Approximations of first-passage times for differentiable processes based on higher-order threshold crossings

Methods for calculating approximations of the first-passage probability of differentiable non-narrow band processes based on higher-order threshold crossings are discussed. Two of them use factorial moments of the number of crossings into the failure region, including a new method based on a Gram-Charlier series expansion of the distribution of the number of exits. Several numerical schemes for the evaluation of factorial moments are investigated. The methods are studied for three examples. The examples show that the Gram-Charlier series expansion converges faster towards the exact solution than Riceʹs “in- and exclusion” series. However, it is difficult to quantify the error made by the proposed method. Further, it is shown that for engineering applications, the Poisson assumption as modified by Ditlevsen such that the initial conditions are taken into account, provides excellent results in almost all cases.