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Lévy processes are useful tools for analysis and modeling of jump‐diffusion processes. Such processes are commonly used in the financial and physical sciences. One approach to building new Lévy processes is through subordination, or a random time change. In this work, we discuss and examine a type of multiply subordinated Lévy process model that we term a deep variance gamma (DVG) process, including estimation and inspection methods for selecting the appropriate level of subordination given data. We perform an extensive simulation study to identify situations in which different subordination depths are identifiable and provide a rigorous theoretical result detailing the behavior of a DVG process as the levels of subordination tend to infinity. We test the model and estimation approach on a data set of intraday 1‐min cryptocurrency returns and show that our approach outperforms other state‐of‐the‐art subordinated Lévy process models.
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Multi-dimensional Lévy Processes for Multiple Dependent Degradation Processes in Lifetime Analysis
The analysis of multiple dependent degradation processes is a challenging research work in the reliability field, especially for complex degradation with random jumps. To integrally handle the jump uncertainties in degradation and the dependence among degradation processes, we construct multi-dimensional Lévy processes to describe multiple dependent degradation processes in engineering systems. The evolution of each degradation process can be modeled by a one-dimensional Lévy subordinator with a marginal Lévy measure, and the dependence among all dimensions can be described by Lévy copulas and the associated multiple-dimensional Lévy measure. This Lévy measure is obtained from all its one-dimensional marginal Lévy measures and the Lévy copula. We develop the Fokker-Planck equations to describe the probability density in stochastic systems. The Laplace transforms of both reliability function and lifetime moments are derived. Numerical examples are used to demonstrate our models in lifetime analysis.
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- Award ID(s):
- 1728321
- PAR ID:
- 10064652
- Date Published:
- Journal Name:
- Proceedings of Industrial and Systems Engineering Research Conference
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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