Levy Processes as Degradation Models

Many degradation models are based on the concept of accumulated damage. Noortwijk [Noo96] points out that in systems subject to shocks, the order in which the damage (i.e. the shocks) occurs is often immaterial so that the random deterioration incurred in equal time intervals forms a set of exchangeable random variables [BS92]. This also implies that the distribution of the degradation incurred is independent of the time scale, i.e. the process has stationary increments. Exchangeable and stationary increments are similar to the stronger properties of stationary and independent increments of Levy processes [Bre68].

The restriction to stationary increments is outweighed by the analytical advantages of using Levy processes. Amongst Levy processes are compound Poisson process, the Wiener process and the gamma process, shot noise process, for which many results are readily available. The Levy-Khinchine decomposition [Bre68, Ch 9,14] expresses any Levy process as the sum of a Wiener process and a jump process with the consequence that any degradation model based on Levy process is either a Wiener process, a jump process or the sum of these two processes. The Wiener process is the only Levy process with continuous sample paths. Thus by insisting that a system whose degradation is continuous is modelled by a Levy process with continuous sample paths restricts the choice to the Wiener process. Similarly, insisting on monotonicity allows only the jump processes within the class of Levy processes [RW94]. Lehmann [Leh04, Leh01] develops a bivariate approach based on the Levy-Khinchine decomposition. Diffusions also arise naturally [Sob87, New91, New98] from the stochastic analogue of the simplest growth laws x'(t} = ax(t}x+1 , namely, dXt = aXx+1dt + (dBt .

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