Markov Additive Processes for Degradation with Jumps under Dynamic Environments

Abstract

We use general Markov additive processes (Markov modulated Lévy processes) to integrally handle the complexity of degradation including internally- and externally-induced stochastic properties with complex jump mechanisms. The background component of the Markov additive process is a Markov chain defined on a finite state space; the additive component evolves as a Lévy subordinator under a certain background state, and may have instantaneous nonnegative jumps occurring at the time the background state switches. We derive the Fokker-Planck equations for such Markov modulated processes, based on which we derive Laplace expressions for reliability function and lifetime moments, represented by the infinitesimal generator matrices of Markov chain and the Lévy measure of Lévy subordinator. The superiority of our models is their flexibility in modeling degradation data with jumps under dynamic environments. Numerical experiments are used to demonstrate that our general models perform well.