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Bayesian prediction for a jump diffusion process – With application to crack growth in fatigue experiments

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  • Hermann, Simone
  • Ickstadt, Katja
  • Müller, Christine H.

Abstract

In many fields of technological developments, understanding and controlling material fatigue is an important point of interest. This article is concerned with statistical modeling of the damage process of prestressed concrete under low cyclic load. A crack width process is observed which exhibits jumps with increasing frequency. Firstly, these jumps are modeled using a Poisson process where two intensity functions are presented and compared. Secondly, based on the modeled jump process, a stochastic process for the crack width is considered through a stochastic differential equation (SDE). It turns out that this SDE has an explicit solution. For both modeling steps, a Bayesian estimation and prediction procedure is presented.

Suggested Citation

  • Hermann, Simone & Ickstadt, Katja & Müller, Christine H., 2018. "Bayesian prediction for a jump diffusion process – With application to crack growth in fatigue experiments," Reliability Engineering and System Safety, Elsevier, vol. 179(C), pages 83-96.
    • Handle: RePEc:eee:reensy:v:179:y:2018:i:c:p:83-96
    DOI: 10.1016/j.ress.2016.08.012
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    References listed on IDEAS

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    1. Casella, Bruno & Roberts, Gareth O., 2011. "Exact Simulation of Jump-Diffusion Processes with Monte Carlo Applications," MPRA Paper 95217, University Library of Munich, Germany.
    2. P. A. W Lewis & G. S. Shedler, 1979. "Simulation of nonhomogeneous poisson processes by thinning," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 26(3), pages 403-413, September.
    3. Bruno Casella & Gareth O. Roberts, 2011. "Exact Simulation of Jump-Diffusion Processes with Monte Carlo Applications," Methodology and Computing in Applied Probability, Springer, vol. 13(3), pages 449-473, September.
    4. Yasutaka Shimizu & Nakahiro Yoshida, 2006. "Estimation of Parameters for Diffusion Processes with Jumps from Discrete Observations," Statistical Inference for Stochastic Processes, Springer, vol. 9(3), pages 227-277, October.
    5. Yu, Jun-Wu & Tian, Guo-Liang & Tang, Man-Lai, 2007. "Predictive analyses for nonhomogeneous Poisson processes with power law using Bayesian approach," Computational Statistics & Data Analysis, Elsevier, vol. 51(9), pages 4254-4268, May.
    6. Yuan, X.-X. & Mao, D. & Pandey, M.D., 2009. "A Bayesian approach to modeling and predicting pitting flaws in steam generator tubes," Reliability Engineering and System Safety, Elsevier, vol. 94(11), pages 1838-1847.
    7. Kay Giesecke & Dmitry Smelov, 2013. "Exact Sampling of Jump Diffusions," Operations Research, INFORMS, vol. 61(4), pages 894-907, August.
    8. Antonio Pievatolo & Fabrizio Ruggeri, 2004. "Bayesian reliability analysis of complex repairable systems," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 20(3), pages 253-264, July.
    9. Simone Hermann & Katja Ickstadt & Christine H. Müller, 2016. "Bayesian prediction of crack growth based on a hierarchical diffusion model," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 32(4), pages 494-510, July.
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    Cited by:

    1. Sun, Xuxue & Cai, Wenjun & Li, Mingyang, 2021. "A hierarchical modeling approach for degradation data with mixed-type covariates and latent heterogeneity," Reliability Engineering and System Safety, Elsevier, vol. 216(C).
    2. Jakobsen, Nina Munkholt & Sørensen, Michael, 2019. "Estimating functions for jump–diffusions," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3282-3318.

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