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

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.