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 引用本文: 陈建兵, 律梦泽. 一类Markov过程的最大绝对值过程概率密度求解的新方法[J]. 力学学报, 2019, 51(5): 1437-1447.
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.
 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.

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

• 摘要: 随机过程或随机系统响应的最大绝对值概率分布往往是科学与工程中关心的重要挑战性问题.本文从理论与数值上进行了Markov过程的时变最大绝对值过程及其概率分布研究.文中,通过引入扩展状态向量,构造了最大绝对值\!-\!-\!状态量联合向量过程,由此将不具有Markov性的最大值过程转化为具有Markov性的向量随机过程.在此基础上,通过最大绝对值\!-\!-\!状态量之间的关系,建立了联合向量过程的转移概率密度函数.进而,结合Chapman-Kolmogorov方程和路径积分方法,提出了最大绝对值概率密度函数求解的数值方法.由此,可以得到Markov过程最大绝对值过程的时变概率密度函数,可进一步用于结构动力可靠度分析等.通过数值算例,验证了本文所提方法的有效性. 该方法有望推广到更一般随机系统的极值分布估计之中.

Abstract: 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.

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