# Analytical explicit solution for the probability density function of a stochastic process-derived random variable – time interval of multiple crossings of the Wiener process and a fixed threshold

### Significance

Problems involving stochastic processes are often associated with engineering applications. When analyzing and solving these problems, it is of great importance to estimate the time intervals between the crossings across a specified threshold within the stochastic process. Pioneering works concentrated mainly on the first-passage problems, that focus on the time exceeding the threshold for the first time. The increasing need for analyzing more complex failure problems such as fatigue failure has compelled researchers to shift to multi-passage problems, which have not been investigated in details. This approach is, however, complex since it involves taking into account the time interval between two adjacent crossings over a specified threshold and the number of crossings.

The Wiener process plays a significant role in stochastic process theory and has been used in numerous fields due to its good analytical properties. To this note, Dr. Zhenhao Zhang from Changsha University of Science and Technology recently explored the multiple-passage problems in Wiener processes. The central focus was a derivation of a probability distribution for a time interval between two adjacent crossings of the Wiener process and a fixed threshold. The work is published in the journal, Mechanical Systems and Signal Processing.

First, the identical nature of time intervals of any two adjacent crossings of the Wiener process was demonstrated. Next, the probability density function of the second passage time across the threshold was successfully derived without any mathematical assumptions. This enabled further derivation of time interval distribution between first- and second-passage time to obtain the desirable explicit analytical expressions. Considering the similarity of the probability distribution of the time intervals, their respective probability density functions were thus generalized. Monte Carlo simulations was good tool for validating the distribution.

The resulting explicit analytical solutions were generally convenient for calculation and applications. The theoretical result of the proposed distribution model was applied to two engineering applications to validate its feasibility. In the first application, the road surface unevenness was analyzed beyond a limit to determine a vehicle’s ride comfort. Based on the Wiener-process-based non-stationary road surface model, the probability distribution presented in this work was used in deriving the probability distribution of the distance interval of a vehicle subjected to extreme road excitation. The excitation information will thus be useful in stochastic response analysis of vehicles. In the second application, the presented approach was suitable for the analysis of the failure probability of electronic products under noise jamming. This was attributed to the invalidity of the common first passage failure mechanism for this case. And so a new time-of-duration-based (or time-interval-based) first passage failure mechanism was presented.

In summary, the research team investigated the time interval produced by multiple crossings of the Wiener process and a fixed threshold. Based on the obtained probability distribution results, its practical applicability was successfully verified in two different engineering applications. Therefore, in a statement to Advances in Engineering, Dr. Zhenhao Zhang mentioned that this work derived an original equation in the field of stochastic process which has potential application in all of fields related to Wiener process, including civil, ocean, mechanical and financial engineering, etc.

Zhenhao Zhang is an associate professor of School of Civil Engineering in Changsha University of Science and Technology, Changsha, China. Dr. Zhang received his Ph.D., M.S. and B.S. degrees from Changsha University of Science and Technology, Changsha, China in 2010, 2006 and 2002, respectively. He worked as a visiting scholar in Department of Civil Engineering at University of Alaska Anchorage, USA from Nov. 2015 to Mar. 2017. He severs as a reviewer for several international journals such as Mechanical Systems and Signal Processing, Journal of Geophysical Research – Oceans, Arabian Journal for Science and Technology, etc. and ever obtained the honor of “outstanding contribution in reviewing” from Mechanical Systems and Signal Processing. He is a member of International Association for Bridge Maintenance and Safety (IABMAS).

A significant body of Dr. Zhang’s research focuses on developing analytical solutions of the probability distributions of stochastic process-derived stochastic variables, which is the key technology of accurate assessment of dynamic reliability. His research interests include structural dynamic reliability, uncertainty quantification and analysis, random vibration of structures, etc. Dr. Zhang won China industry-university-research collaboration innovation achievement award in 2019 which highlights his recognition as a career researcher at high level of excellence.

His publications include a monograph, a high education textbook, five local technical specifications of Hunan province in China and more than sixty journal and conference papers. He has organized several mini-symposia or sessions in high level international conferences in his field. Dr. Zhang gained very early financial support from National Natural Science foundation of China (NSFC) as Principal Investigator which highlights his high level research work.

Email: [email protected]

### Reference

Zhang, Z., Liu, X., Zhang, Y., Zhou, M., & Chen, J. (2020). Time interval of multiple crossings of the Wiener process and a fixed threshold in engineering. Mechanical Systems and Signal Processing, 135, 106389.