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We consider symmetric stable Levy motion time-changed by tempered stable subordinator. This process generalizes the normal inverse Gaussian process without drift term, introduced by Barndorff-Nielsen. The asymptotic tail behavior of the density function of this process and corresponding Levy density is obtained. The governing Fokker–Planck–Kolmogorov equation of the density function of the introduced process in terms of shifted fractional derivative is established. Codifference and asymptotic behavior of the moments are discussed. Further, we also introduce and analyze stable subordinator delayed by tempered stable subordinator.

In this paper, two new estimators based on the spirit of the best linear unbiased estimators are separately developed for homogeneous gamma process. Both estimators can be computed recursively, and have high efficiency. We compare the two new estimators with those of Paroissin (2017) by simulations, and find that ours have smaller biases and mean squared errors.

Remaining useful life (RUL) prediction plays an important role in the operation and health management of hydraulic piston pumps. The inverse Gaussian (IG) process model is a flexible alternative for the RUL prediction of hydraulic piston pumps. However, random effects and measurement errors are not taken into account during the prediction process, which results in inaccurate prediction results. To improve the RUL prediction accuracy of hydraulic piston pumps, this paper proposes an improved IG process model by considering the random effects and measurement errors to describe the wear degradation. The measurement error is statistically dependent on the degradation state of the actual degradation process. Monte Carlo integration and the expectation maximization (EM) algorithm are further developed to estimate the parameters. Finally, the accuracy and effectiveness of the proposed model are demonstrated through two case studies. The results show that the improved IG process model can improve the RUL prediction accuracy.

We derive the governing equation of the Tempered Stable Subordinator (hereafter TSS), which generalizes the space-fractional differential equation satisfied by the law of the α-stable subordinator itself. This equation is expressed in terms of the shifted fractional derivative of order α ? ( 0 , 1 ) coinciding with the stability parameter. We then generalize this equation by introducing a time-fractional derivative of order β ? ( 0 , 1 ) (resp. 1 / β 1 ) and we prove that it is satisfied by the law of a TSS time-changed by the inverse of a β-stable subordinator (resp. by the stable subordinator itself). The corresponding processes can therefore be called "fractional TS processes". Finally we provide fractional extensions of the relativistic stable processes, which we define as a Brownian motion with a random time argument represented by independent fractional TS processes of order β (resp. 1 / β ).

We consider time-changed Poisson processes, and derive the governing difference-differential equations (DDE) these processes. In particular, we consider the time-changed Poisson processes where the the time-change is inverse Gaussian, or its hitting time process, and discuss the governing DDE's. The stable subordinator, inverse stable subordinator and their iterated versions are also considered as time-changes. DDE's corresponding to probability mass functions of these time-changed processes are obtained. Finally, we obtain a new governing partial differential equation for the tempered stable subordinator of index $0, Comment: 18 pages

In this paper we extend the method of Bernstein using some concepts of differential geometry and prove the existence of a partial differential equation which when solved under suitable boundary conditions leads to a density of a bivariate inverse Gaussian process.