What is it about?

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.

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Why is it important?

This paper has demonstrated that the time intervals of any two adjacent crossings of the Wiener process are identically distributed, and provide the explicit analytical solution of the probability distribution function of the time interval between the first-passage time and the second-passage time (or called crossing-threshold duration).

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This page is a summary of: Time interval of multiple crossings of the Wiener process and a fixed threshold in engineering, Mechanical Systems and Signal Processing, January 2020, Elsevier, DOI: 10.1016/j.ymssp.2019.106389.
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