Phase Type Modelling for Approximation of the Geometric Brownian Motion

Abstract

A geometric Brownian motion (GBM) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion with drift. The GBM allows to model degradation where the increase in the degradation level depends on the current degradation level. Many physical degradation phenomena are characterized by laws reflecting this fact. In this paper a continuous-time Markov chain is used to approximate the GBM. There are two main reasons for this, firstly the available data only provides discretized state information. Further a continuous-time Markov Chain is often more attractive to work with than the stochastic processes where the state variable is continuous. A weakness of the continuous-time Markov Chain to model continuous degradation is that sojourn times could not realistically be modelled by the exponential distribution. In this paper we therefore allow the sojourn times to have non-Markovian properties. To accomplish this we use phase type distributions where each `main state" is modelled by a continuous phase-type distribution. Since the Brownian motion allows for "negative increments" we also implement a mechanism allowing to move in both direction in the Markov chain. Numerical results are given to demonstrate the validity of the approximation. Case studies are provided to support useful residual lifetime (RUL), and determination of optimal inspection and maintenance regime under a predictive maintenance regime. Markov chain models are frequently used for RUL prediction where a fixed failure limit exist. In many real case situations there are no fixed failure limit, and therefore a model with a random failure limit is derived to support more realistic modelling.