Self-exciting jump processes and their asymptotic behaviour

Dahl, Kristina Rognlien; Eyjolfsson, Heidar

doi:10.1080/17442508.2022.2028789

Cited by 2 publications

(6 citation statements)

References 14 publications

(24 reference statements)

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“…Infinite activity processes, such as the gamma process, are used to model deterioration caused by gradual wear. Dahl and Eyjolfsson (2021) show that the selfexciting model presented in Section 2 is a finite activity model. However, they also prove that when the model parameters are chosen in a suitable way, and then passed to a limit, we get an infinite activity process in the limit.…”

Section: Finite and Infinite Activitymentioning

confidence: 91%

“…Our definition of self-exciting processes, follows Eyjolfsson and Tjøstheim (2018), and the following presentation of the framework can also be found in Dahl and Eyjolfsson (2021), but is included here for completeness. The self-exciting process is a counting process, counting the number of shocks that have happened at any particular time.…”

Section: Self-exciting Jump Processesmentioning

confidence: 99%

“…This example can also be found in Dahl and Eyjolfsson (2021), but is included here for completeness.…”

Section: Self-exciting Jump Processesmentioning

confidence: 99%

“…Define Λ(t) := t 0 λ(u)du. Then, Dahl and Eyjolfsson Dahl and Eyjolfsson (2021) show that a sufficient condition for the SDE-driven selfexciting process (as defined in Section 2) to be of finite activity is that for all times T > 0, Λ(T ) < ∞ holds almost surely.…”

Section: Finite and Infinite Activitymentioning

confidence: 99%

“…As seen in Dahl and Eyjolfsson (2021), the selfexciting model of this paper is essentially a finite activity jump process model (like the CPP): In bounded time intervals the process only produces a finite numbers of jumps. However, Dahl and Eyjolfsson (2021) found that when the model parameters are chosen in a suitable way, and then passed to a limit, we get an infinite activity process in the limit. This can be thought of as an infinite activity analogue of the self-exciting process.…”

Section: Introductionmentioning

confidence: 99%

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Self-Exciting Jump Processes as Deterioration Models

Dahl¹,

Eyjolfsson²

2021

Proceedings of the 31st European Safety and Reliability Conference (ESREL 2021)

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Several different approaches to modelling stochastic deterioration for optimising maintenance have been suggested in the reliability literature. These include component lifetime distributions, which have the disadvantage of being binary, in the sense of only telling whether the component has failed or not. Failure rate functions model ageing in a more satisfactory way than lifetime distributions. However, failure rates cannot be observed for a single component, and are therefore not tractable in practical applications. To mitigate this, a theory for modeling deterioration via stochastic processes developed. Various processes have been suggested, such as Brownian motion with drift and compound Poisson processes (CPP) for modeling usage and damage from sporadic shocks and gamma processes to model gradual ageing. However, none of these processes are able to capture jump clustering. To allow for clustering of jumps (failure events), we suggest an alternative approach in this paper: To use self-exciting jump processes to model stochastic deterioration of components in a system where there may be clustering effects in the degradation. Self-exciting processes excite their own intensity, so large shocks are likely to be followed by another shock within a short period of time. Furthermore, self-exciting processes may have both finite and infinite activity. Therefore, we suggest that these processes can be used to model degradation both by sporadic shocks and by gradual wear. We illustrate the use of self-exciting degradation with several numerical examples. In particular, we use Monte Carlo simulation to estimate the expected lifetime of a component with self-exciting degradation. As an illustration, we also estimate the lifetime of a bridge system with independent components with identically distributed self-exciting degradation.

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Section: Finite and Infinite Activitymentioning

confidence: 91%

Section: Self-exciting Jump Processesmentioning

confidence: 99%

“…This example can also be found in Dahl and Eyjolfsson (2021), but is included here for completeness.…”

Section: Self-exciting Jump Processesmentioning

confidence: 99%

Section: Finite and Infinite Activitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%