a b s t r a c tWe consider the first-exit time of a tempered β-stable subordinator, also called inverse tempered stable (ITS) subordinator. An integral form representation and a series representation for the density of the ITS subordinator are obtained. The asymptotic behavior of the q-th order moments of the ITS subordinator are investigated. The limiting form of the ITS density and its k-th order derivatives are derived as the space variable x → 0 + . Finally, the governing pde of the ITS density is also obtained.

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 < β < 1, when β is a rational number. We then use this result to obtain the governing DDE for the mass function of Poisson process time-changed by tempered stable subordinator. Our results extend and complement the results in Baeumer et al. [3] and Beghin et al. [4] in several directions.

In recent years subordinated processes have been widely considered in the literature. These processes not only have wide applications but also have interesting theoretical properties. In this paper we consider fractional Brownian motion (FBM) time-changed by two processes, tempered stable and inverse tempered stable. We present main properties of the subordinated FBM such as long range dependence and associated fractional partial differential equations for the probability density functions. Moreover, we present how to simulate both subordinated processes.

Abstract. Many real time-series exhibit behavior adequate to long range dependent data. Additionally very often these time-series have constant time periods and also have characteristics similar to Gaussian processes although they are not Gaussian. Therefore there is need to consider new classes of systems to model these kind of empirical behavior. Motivated by this fact in this paper we analyze two processes which exhibit long range dependence property and have additional interesting characteristics which may be observed in real phenomena. Both of them are constructed as the superposition of fractional Brownian motion (FBM) and other process. In the first case the internal process, which plays role of the time, is the gamma process while in the second case the internal process is its inverse. We present in detail their main properties paying main attention to the long range dependence property. Moreover, we show how to simulate these processes and estimate their parameters. We propose to use a novel method based on rescaled modified cumulative distribution function for estimation of parameters of the second considered process. This method is very useful in description of rounded data, like waiting times of subordinated processes delayed by inverse subordinators. By using the Monte Carlo method we show the effectiveness of proposed estimation procedures.Keywors: subordination, gamma process, inverse gamma process, simulation, estimation

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