A new approach for time-variant probability density function of the maximal value of stochastic dynamical systems

Abstract

The extreme value distribution (EVD) of stochastic processes is an important but still challenging problem for the determination of reliability function and distribution of first excursion time in various science and engineering fields. In the present paper, a new method to evaluate the time-variant probability density function (PDF) of the maximal value of a Markov process or Markov vector process is proposed. In this method, a joint maximum-state vector process is constructed by combining the maximal value process (MVP) and its underlying Markov process. The Markov property of the joint maximum-state vector process is rigorously proved. Incorporated with the Itô stochastic differential equation (SDE) governing the underlying Markov process, a numerical method is developed based on the Chapman-Kolmogorov equation. In particular, the short-time transition probability density function of the joint maximum-state process in the path integral solution (PIS) is derived. The detailed algorithms for the proposed method in two- and three-dimensional cases, respectively, are elaborated. Several examples are illustrated, demonstrating the effectiveness of the proposed method. Problems to be further studied are also discussed.