Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse

Maheshwari, A.; Vellaisamy, P.

doi:10.1007/s10959-017-0797-6

Cited by 26 publications

(13 citation statements)

References 30 publications

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“…We know that LT of pth order moment of inverse subordinator {E f (t)} t≥0 is given by (see [16]) Proof. LetM(s) be the LT of E[E…”

Section: ũ(S)mentioning

confidence: 99%

Subordinated compound Poisson processes of order k

Sengar¹,

Upadhye²

2020

Modern Stochastics: Theory and Applications

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In this article, the compound Poisson process of order k (CPPoK) is introduced and its properties are discussed. Further, using mixture of tempered stable subordinators (MTSS) and its right continuous inverse, the two subordinated CPPoK with various distributional properties are studied. It is also shown that the space and tempered space fractional versions of CPPoK and PPoK can be obtained, which generalize the process defined in [Statist. Probab. Lett. 82 (2012), 852-858]. Keywords Compound Poisson process of order k, mixture of tempered stable subordinators, martingale characterization 2010 MSC 60G51, 60G48

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“…We know that LT of pth order moment of inverse subordinator {E f (t)} t≥0 is given by (see [16]) Proof. LetM(s) be the LT of E[E…”

Section: ũ(S)mentioning

confidence: 99%

Subordinated compound Poisson processes of order k

Sengar¹,

Upadhye²

2020

Modern Stochastics: Theory and Applications

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“…To get the expression of variance of the TCPPoK-I, we can put s = t in the Part (ii). [24] for more details) is greater than 1. Hence, we conclude that TCPPoK-I exhibits overdispersion.…”

Section: Example 33 (Poisson-inversementioning

confidence: 99%

“…. t n , using the Algorithm 2-5 of [24] for respective subordinator (inverse subordinator). (d) Using the values W (t i ), 1 ≤ i ≤ n, generated in Step (c), as time points, compute the number of events of the PPoK {N (k) (W (t i ))}, 1 ≤ i ≤ n, using Algorithm 1.…”

Section: Simulationmentioning

confidence: 99%

Time-changed Poisson processes of order k

Sengar

Maheshwari

Upadhye

2019

Stochastic Analysis and Applications

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In this article, we study the Poisson process of order k (PPoK) time-changed with an independent Lévy subordinator and its inverse, which we call respectively, as TCPPoK-I and TCPPoK-II, through various distributional properties, long-range dependence and limit theorems for the PPoK and the TCPPoK-I. Further, we study the governing difference-differential equations of the TCPPoK-I for the case inverse Gaussian subordinator. Similarly, we study the distributional properties, asymptotic moments and the governing difference-differential equation of TCPPoK-II. As an application to ruin theory, we give a governing differential equation of ruin probability in insurance ruin using these processes. Finally, we present some simulated sample paths of both the processes.2010 Mathematics Subject Classification. 60G55; 60G51.

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“…For brief discussion and other possible definitions of long-range dependence in various settings see Heyde & Yang (1997). Long-range dependence property in the form of Definition 5.1 was first used in Maheshwari & Vellaisamy (2016) and Maheshwari & Vellaisamy (2017). Moreover, equivalent form was used in Leonenko et al (2013a).…”

Section: Long-range Dependence For Dcar(p) Processmentioning

confidence: 99%