Lévy-driven non-Gaussian Ornstein–Uhlenbeck processes for degradation-based reliability analysis

ABSTRACT We use Lévy subordinators and non-Gaussian Ornstein–Uhlenbeck processes to model the evolution of degradation with random jumps. The superiority of our models stems from the flexibility of such processes in the modeling of stylized features of degradation data series such as jumps, linearity/nonlinearity, symmetry/asymmetry, and light/heavy tails. Based on corresponding Fokker–Planck equations, we derive explicit results for the reliability function and lifetime moments in terms of Laplace transforms, represented by Lévy measures. Numerical experiments are used to demonstrate that our general models perform well and are applicable for analyzing a large number of degradation phenomena. More important, they provide us with a new methodology to deal with multi-degradation processes under dynamicenvironments.

[1]  Michael E. Cholette,et al.  Degradation modeling and monitoring of machines using operation-specific hidden Markov models , 2014 .

[2]  R. Maller,et al.  Ornstein–Uhlenbeck Processes and Extensions , 2009 .

[3]  Kazimierz Sobczyk.  Stochastic models for fatigue damage of materials , 1989 .

[4]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[5]  Jinqiao Duan,et al.  Fractional Fokker-Planck Equation for Nonlinear Stochastic Differential Equations Driven by Non-Gaussian Levy Stable Noises , 1999, math/0409486.

[6]  F. Klebaner.  Introduction To Stochastic Calculus With Applications , 1999 .

[7]  A. W. Marshall,et al.  Shock Models and Wear Processes , 1973 .

[8]  L. Brancik.  Numerical Inversion of Two-Dimensional Laplace Transforms Based on Partial Inversions , 2007, 2007 17th International Conference Radioelektronika.

[9]  H. Kramers.  Brownian motion in a field of force and the diffusion model of chemical reactions , 1940 .

[10]  Hideitsu Hino,et al.  The YUIMA Project: A Computational Framework for Simulation and Inference of Stochastic Differential Equations , 2014 .

[11]  Mark E. Oxley,et al.  Reliability of manufacturing equipment in complex environments , 2013, Ann. Oper. Res..

[12]  Kwok-Leung Tsui,et al.  Condition monitoring and remaining useful life prediction using degradation signals: revisited , 2013 .

[13]  N. Shephard,et al.  Econometric analysis of realized volatility and its use in estimating stochastic volatility models , 2002 .

[14]  G. A. Whitmore,et al.  Failure Inference From a Marker Process Based on a Bivariate Wiener Model , 1998, Lifetime data analysis.

[15]  O. Aalen,et al.  Survival Models Based on the Ornstein-Uhlenbeck Process , 2004, Lifetime data analysis.

[16]  Loon Ching Tang,et al.  Degradation-Based Burn-In Planning Under Competing Risks , 2012, Technometrics.

[17]  Maurizio Guida,et al.  A New Class of Markovian Processes for Deteriorating Units With State Dependent Increments and Covariates , 2015, IEEE Transactions on Reliability.

[18]  Ting Gao,et al.  Mean Exit Time and Escape Probability for a Tumor Growth System under Non-Gaussian noise , 2011, Int. J. Bifurc. Chaos.

[19]  Dustin G. Mixon,et al.  On a Markov‐modulated shock and wear process , 2009 .

[20]  Qiong Wu,et al.  Degradation reliability modeling based on an independent increment process with quadratic variance , 2016 .

[21]  Dustin G. Mixon,et al.  Availability of periodically inspected systems with Markovian wear and shocks , 2006, Journal of Applied Probability.

[22]  S. I. Denisov,et al.  Generalized Fokker-Planck equation: Derivation and exact solutions , 2008, 0808.0274.

[23]  Jian Yang,et al.  Constrained hierarchical modeling of degradation data in tissue-engineered scaffold fabrication , 2016 .

[24]  Neil Shephard,et al.  Integrated OU Processes and Non‐Gaussian OU‐based Stochastic Volatility Models , 2003 .

[25]  G. Jongbloed,et al.  Parametric Estimation for Subordinators and Induced OU Processes , 2006 .

[26]  Jeffrey P. Kharoufeh,et al.  Explicit results for wear processes in a Markovian environment , 2003, Oper. Res. Lett..

[27]  Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian Lévy processes , 2012, 1202.2563.

[28]  David Applebaum,et al.  Lévy Processes and Stochastic Calculus by David Applebaum , 2009 .

[29]  Ward Whitt,et al.  Numerical Inversion of Laplace Transforms of Probability Distributions , 1995, INFORMS J. Comput..

[30]  佐藤 健一.  Lévy processes and infinitely divisible distributions , 2013 .

[31]  Oldrich A. Vasicek.  An equilibrium characterization of the term structure , 1977 .

[32]  N. Shephard,et al.  Basics of Levy processes , 2012 .

[33]  R. Mazo.  On the theory of brownian motion , 1973 .

[34]  Guang Jin,et al.  Reliability Demonstration for Long-Life Products Based on Degradation Testing and a Wiener Process Model , 2014, IEEE Transactions on Reliability.

[35]  Loon Ching Tang,et al.  Semiparametric Estimation of Gamma Processes for Deteriorating Products , 2014, Technometrics.

[36]  H. Risken.  Fokker-Planck Equation , 1996 .

[37]  M. Abdel-Hameed.  A Gamma Wear Process , 1975, IEEE Transactions on Reliability.

[38]  W. Schoutens.  Lévy Processes in Finance: Pricing Financial Derivatives , 2003 .

[39]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[40]  Liu Xiao,et al.  Criticality measures for components with multi-dimensional degradation , 2012 .

[41]  Jan M. van Noortwijk,et al.  A survey of the application of gamma processes in maintenance , 2009, Reliab. Eng. Syst. Saf..

[42]  R. Schilling.  Financial Modelling with Jump Processes , 2005 .

[43]  Erhan Çinlar,et al.  SHOCK AND WEAR MODELS AND MARKOV ADDITIVE PROCESSES , 1977 .

[44]  Mohamed Abdel-Hameed,et al.  Life Distribution Properties of Devices Subject to a Lévy Wear Process , 1984, Math. Oper. Res..

[45]  Ming J. Zuo,et al.  Multistate degradation and supervised estimation methods for a condition-monitored device , 2014 .

[46]  David W. Coit,et al.  Life distribution analysis based on Lévy subordinators for degradation with random jumps , 2015 .