Time interval of multiple crossings of the Wiener process and a fixed threshold in engineering

In analyzing the practical engineering problems involving stochastic processes, a key component is the estimation of time intervals between the crossings across a specified threshold. However, the former research works are mainly focusing on first-passage problem, the multiple-passage problem, more importantly, have not been investigated in details. In this paper, the multiple-passage problem associated with the Wiener process is studied. The work is focusing on the derivation of probability distribution for the time interval between two adjacent crossings of a Wiener process. First, it is demonstrated that the time intervals of any two adjacent crossings of the Wiener process are identically distributed. The probability density function of the second-passage time of a Wiener process across a threshold is derived. This is then implemented in the derivation of distribution of the time interval between the first-passage time and the second-passage time. Based on the derivation of the joint probability density function of the two-dimensional random vector consisting of the first-passage time and the second-passage time, the explicit analytical expression of the probability distribution function of the time interval between the first-passage time and the second-passage time are obtained finally. Since the probability distribution of each time interval is identical, the probability density function of each time interval is thus generalized. To demonstrate the proposed distribution model, the theoretical result is applied in the analysis of road surface unevenness beyond a limit and the failure probability analysis of electronic products under noice jamming, which shows that the developed approach has important applications in different fields of engineering. (C) 2019 Elsevier Ltd. All rights reserved.