EI、Scopus 收录
中文核心期刊
Chen Jianbing, Lü Mengze. A NEW METHOD FOR THE PROBABILITY DENSITY OF MAXIMUM ABSOLUTE VALUE OF A MARKOV PROCESS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(5): 1437-1447. DOI: 10.6052/0459-1879-19-104
Citation: Chen Jianbing, Lü Mengze. A NEW METHOD FOR THE PROBABILITY DENSITY OF MAXIMUM ABSOLUTE VALUE OF A MARKOV PROCESS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(5): 1437-1447. DOI: 10.6052/0459-1879-19-104

A NEW METHOD FOR THE PROBABILITY DENSITY OF MAXIMUM ABSOLUTE VALUE OF A MARKOV PROCESS

  • Received Date: April 24, 2019
  • The probability distribution of maximum absolute value of stochastic processes or responses of stochastic systems is a significant problem in various science and engineering fields. In the present paper, theoretical and numerical studies on the time-variant extreme value process and its probability distribution of Markov process are performed. An augmented vector process is constructed by combining the maximum absolute value process and its underlying Markov process. Thereby, the non-Markov maximum absolute value process is converted to a Markov vector process. The transitional probability density function of the augmented vector process is derived based on the relationship between the maximum absolute process and the underlying state process. Further, by incorporating the Chapman-Kolmogorov equation and the path integral solution method, the numerical methods for the probability density function of the maximum absolute value is proposed. Consequently, the time-variant probability density function of the maximum absolute of a Markov process can be obtained, and can be applied, e.g., to the evaluation of dynamic reliability of engineering structures. Numerical examples are illustrated, demonstrating the effectiveness of the proposed method. The proposed method is potentially to be extended for the estimation of extreme value distribution of more general stochastic systems.
  • [1] Arrechi FT . Transition Phenomena in Nonlinear Optics. Berlin: Springer, 1981
    [2] Bras RL . Hydrology: An Introduction to Hydrological Science. Reading: Addison-Wesley, 1990
    [3] 胡岗 . 随机力与非线性系统. 上海: 上海科技教育出版社, 1994
    [3] ( Hu Gang. Stochastic Forces and Nonlinear Systems. Shanghai: Shanghai Scientific and Technological Education Publishing House, 1994 (in Chinese))
    [4] ?ksendal B. Stochastic Differential Equations. 5th Ed. Berlin: Springer-Verlag, 1998
    [5] Hull JC. Option , Futures, Other Derivatives. 4th Ed. Upper Saddle River: Prentice-Hall, 2000
    [6] Li J, Chen JB . Stochastic Dynamics of Structure. Singapore: John Wiley & Sons (Asia) Pte Ltd, 2009
    [7] 苏成, 徐瑞 . 非平稳随机激励下结构体系动力可靠度时域解法. 力学学报, 2010,42(3):512-520
    [7] ( Su Cheng, Xu Rui . Time-domain method for dynamic reliability of structural systems subjected to non-stationary random excitations. Chinese Journal of Theoretical and Applied Mechanics, 2010,42(3):512-520 (in Chinese))
    [8] Fisher RA, Tippett LHC . Limiting forms of the frequency distribution of the largest and smallest member of a sample. Mathematical Proceedings of the Cambridge Philosophical Society, 1928,24(2):180-190
    [9] Gumbel EJ. Statistics of Extremes. New York: Columbia University Press, 1958
    [10] Coles S . An Introduction to Statistical Modeling of Extreme Values. Springer, 2001
    [11] Powell A . On the fatigue failure of structures due to vibration excited by random pressure fields. The Journal of the Acoustical Society of America, 1958,30(2):1130-1135
    [12] Dudley RM. Real Analysis and Probability. Cambridge: The Press Syndicate of the University of Cambridge, 1989
    [13] Klebaner FC. Introduction to Stochastic Calculus with Application. 2nd Ed. London: Imperial College Press, 2005
    [14] Redner S. A Guide to First-Passage Processes. Cambridge: Cambridge University Press, 2001
    [15] 陈建兵, 李杰 . 非线性随机结构动力可靠度的密度演化方法. 力学学报, 2004,36(2):196-201
    [15] ( Chen Jianbing, Li Jie . The probability density evolution method for dynamic reliability assessment of nonlinear stochastic structures. Chinese Journal of Theoretical and Applied Mechanics, 2004,36(2):196-201 (in Chinese))
    [16] Monili A, Talkner P, Katul GG , et al. First passage time statistics of Brownian motion with purely time dependent drift and diffusion. Physica A - Statistical Mechanics & Its Applications, 2011,390:1841-1852
    [17] Kou S, Zhong H . First-passage times of two-dimensional Brownian motion. Advanced Applied Probability, 2016,48:1045-1060
    [18] Li J, Chen JB . The principle of preservation of probability and the generalized density evolution equation. Structural Safety, 2008,30:65-77
    [19] 陈建兵, 张圣涵 . 非均布随机参数结构非线性响应的概率密度演化. 力学学报, 2014,46(1):136-144
    [19] ( Chen Jianbing, Zhang Shenghan . Probability density evolution analysis of nonlinear response of structures with non-uniform random parameters. Chinese Journal of Theoretical and Applied Mechanics, 2014,46(1):136-144 (in Chinese))
    [20] Chen JB, Li J . The extreme value distribution and dynamic reliability analysis of nonlinear structures with uncertain parameters. Structural Safety, 2007,29:77-93
    [21] Mannella R, Palleschi V . Fast and precise algorithm for computer simulation of stochastic differential equations. Physical Review A, 1989,40(6):3381-3386
    [22] Honeycutt RL . Stochastic Runge-Kutta algorithms. I. White noise. Physical Review A, 1992,45(2):600-603
    [23] Honeycutt RL . Stochastic Runge-Kutta algorithms. II. Colored noise. Physical Review A, 1992,45(2):604-610
    [24] Higham DJ . An algorithmic introduction to numerical simulation of stochastic differential equations. Society for Industrial and Applied Mathematics, 2001,43(3):525-546
    [25] 朱位秋. 随机振动. 北京:科学出版社, 1992
    [25] ( Zhu Weiqiu. Random Vibration. Beijing: Science Press, 2003)
    [26] Gardiner CW . Handbook of Stochastic Methods for Physics. 2nd Ed. Berlin: Springer-Verlag, 1985
    [27] 徐伟 . 非线性随机动力学的若干数值方法及应用. 北京: 科学出版社, 2013
    [27] ( Xu Wei. Numerical Analysis Methods for Stochastic Dynamical System. Beijing: Science Press, 2013 (in Chinese))
    [28] Risken H . The Fokker-Planck Equation. 2nd Ed. Berlin: Springer-Verlag, 1989
    [29] Chen JB, Lyu MZ . A new approach for the time-variant probability density function of the maximum value of a Markov process. Journal of Computational Physics, 2019 ( under review)
    [30] Lyu MZ, Chen JB, Pirrotta A . A novel method based on augmented Markov vector process for the time-variant extreme value distribution of stochastic dynamical systems enforced by Poisson white noise. Communications in Nonlinear Science and Numerical Simulation, 2019 ( accepted)
    [31] Er GK . Exponential closure method for some randomly excited non-linear systems. International Journal of Non-Linear Mechanics, 2000,35:69-78
    [32] 朱位秋 . 非线性随机动力学与控制---Hamilton理论体系框架. 北京: 科学出版社, 2003
    [32] ( Zhu Weiqiu. Nonlinear Stochastic Dynamics and Control---Hamiltonian Formulation. Beijing: Science Press, 2003 (in Chinese))
    [33] Chen JB, Yuan SR . Dimension Reduction of the FPK Equation via an Equivalence of Probability Flux for Additively Excited Systems. Journal of Engineering Mechanics, 2014,140(11):04014088
    [34] Chen JB, Rui ZM . Dimension-reduced FPK equation for additive white-noise excited nonlinear structures. Probabilistic Engineering Mechanics, 2018,53:1-13
    [35] 芮珍梅, 陈建兵 . 加性非平稳激励下结构速度响应的FPK方程降维. 力学学报, 2019,51(3):922-931
    [35] ( Rui Zhenmei, Chen Jianbing . Dimension reduction of FPK equation for velocity response analysis of structures subjected to additive nonstationary excitations. Chinese Journal of Theoretical and Applied Mechanics, 2019,51(3):922-931 (in Chinese))
  • Related Articles

    [1]Lu Xiyun. RESEARCH ON MULTI-PROCESS PROBLEMS IN EXTREME FLOWS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2025, 57(4): 1028-1030. DOI: 10.6052/0459-1879-25-145
    [2]Fu Kangqi, Zhang Lerong, Li Qingjun, Deng Zichen, Wu Zhigang, Jiang Jianping. DYNAMICS AND ATTITUDE CONTROL OF THE ASSEMBLY PROCESS OF ULTRA-LARGE MULTI-MODULE STRUCTURES[J]. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(2): 446-459. DOI: 10.6052/0459-1879-23-289
    [3]Sun Yibo, Wei Sha, Ding Hu, Chen Liqun. STOCHASTIC DYNAMIC RESPONSE ANALYSIS OF PIPE CONVEYING FLUID BASED ON THE PATH INTEGRAL METHOD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(6): 1371-1381. DOI: 10.6052/0459-1879-23-032
    [4]Li Zhijing, Cao Zhixian, Hu Peng, Gareth Pender. MULTIPLE TIME SCALES OF AEOLIAN AND FLUVIAL PROCESSES AND DEPTH-AVERAGED INTEGRAL MODELLING[J]. Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(2): 158-163. DOI: 10.6052/0459-1879-12-311
    [5]Zhang Hongping Sun Chengwei Li Mu Zhao Jianheng. Backward integration method in data processing of quasi-isentropic compression experiment[J]. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(1): 105-111. DOI: 10.6052/0459-1879-2011-1-lxxb2010-053
    [6]Hillslope soil erosion process model for natural rainfall events[J]. Chinese Journal of Theoretical and Applied Mechanics, 2008, 40(3). DOI: 10.6052/0459-1879-2008-3-2006-329
    [7]Shoujie He, Jing Ha, Xuechen Li, Qing Li, Long Wang. Theory of the dynamics process of the conical bubble sonoluminescence[J]. Chinese Journal of Theoretical and Applied Mechanics, 2007, 23(6): 727-731. DOI: 10.6052/0459-1879-2007-6-2007-131
    [8]THE MULTI-RITZ-VECTOR METHOD IN GENERALIZED EIGENVALUE PROBLEMS[J]. Chinese Journal of Theoretical and Applied Mechanics, 1999, 31(5): 585-595. DOI: 10.6052/0459-1879-1999-5-1995-069
    [9]MATHEMATICAL MODELLING OF 3-DIMENSIONAL TURBULENT FLOW BY VORTICITY-VECTOR POTENTIAL METHOD[J]. Chinese Journal of Theoretical and Applied Mechanics, 1991, 23(2): 157-164. DOI: 10.6052/0459-1879-1991-2-1995-822
    [10]样条积分方程法分析弹塑性板弯曲[J]. Chinese Journal of Theoretical and Applied Mechanics, 1990, 22(2): 241-245. DOI: 10.6052/0459-1879-1990-2-1995-940

Catalog

    Article Metrics

    Article views (1527) PDF downloads (120) Cited by()
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return