Closed-Form Solutions for the Probability Distribution of Time-Variant Maximal Value Processes for Some Classes of Markov Processes

Abstract

The time-variant maximal value process (MVP) of a Markov process has significant applications in various science and engineering fields. In the present paper, the closed-form solutions for the probability distribution of the time-variant MVP for some classes of Markov process are studied. For general continuous Markov processes, a unified Volterra integral equation governing the evolution of the cumulative distribution functions (CDFs) of the time-variant MVP of a Markov process is established for the first time. Closed-form or numerical solutions for MVP of some special continuous Markov processes are derived according to this equation. For the compound Poisson process, which is a discontinuous Markov process, the closed-form solution of concentrated probability of the time-variant MVP at zero point is given analytically. Finally, several examples are illustrated as case studies of these theoretical results, demonstrating the effectiveness of the results. Though the analytical results are now only applicable to one-dimensional Markov process, it provides at least some benchmark results for the checking of future possible analytical or numerical methods for the probability density function (PDF) of MVP of more general and high-dimensional Markov process. Further, it provides insights that might stimulate more sophisticated results in the future.