The Folded Wiener Process for Non-negative Non-monotone Degradation Processes

Abstract

The Wiener process is very largely and successfully adopted for modelling degradation phenomena that can exhibit negative increments. However, so far as the observed degradation phenomenon is intrinsically non-negative, as generally occurs, its use may lead to incongruous results, unless (based on the parameter setting) the probability that the considered Wiener process takes negative values is negligible for any time of interest. To overcome such a limitation of the Wiener process, a new Markovian degradation process, called the folded Wiener process, is here proposed that can be used to describe intrinsically non-negative, non-monotone, degradation phenomena. In fact, under the folded Wiener process, the degradation increment over future time intervals is constrained to be no less than the opposite of the current degradation level, so that the total amount of degradation accumulated up to any time \( t \) is necessarily non-negative. The main features and properties of the folded Wiener process are then presented and discussed, and closed form expressions for the mean and variance functions of the process are also provided. Finally, to show the feasibility of the proposed folded Wiener process, an applicative example is developed by using real data.KeywordsNon-monotone degradation processNon-negative degradationFolded normal distribution