State dependent versions of the space-time fractional poisson process
-
Kuldeep Kumar Kataria
and
Palaniappan Vellaisamy
Abstract
In this paper, we introduce and study two counting processes by considering state dependency on the order of fractional derivative as well as on the exponent of backward shift operator involved in the governing difference-differential equations of the state probabilities of space-time fractional Poisson process. The Adomian decomposition method is employed to obtain their state probabilities and then their Laplace transforms are evaluated. Also, the compound versions of these state dependent models are studied and the corresponding governing fractional integral equations of their state probabilities are obtained.
Acknowledgements
The authors would like to thank the anonymous reviewers for their insightful comments and suggestions that have led to improvements.
References
[1] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic, Dordrecht (1994).10.1007/978-94-015-8289-6Search in Google Scholar
[2] M.S. Alrawashdeh, J.F. Kelly, M.M. Meerschaert, H.P. Scheffler, Applications of inverse tempered stable subordinators. Comput. Math. Appl. 73, No 6 (2017), 892–905; 10.1016/j.camwa.2016.07.026.Search in Google Scholar
[3] L. Beghin, C. Macci, Fractional discrete processes: compound and mixed Poisson representations. J. Appl. Probab. 51, No 1 (2014), 19–36; 10.1239/jap/1395771411.Search in Google Scholar
[4] L. Beghin, E. Orsingher, Fractional Poisson processes and related planar random motions. Electron. J. Probab. 14, No 61 (2009), 1790–1827; 10.1214/EJP.v14-675.Search in Google Scholar
[5] L. Beghin, E. Orsingher, Poisson-type processes governed by fractional and higher-order recursive differential equations. Electron. J. Probab. 15, No 22 (2010), 684–709; 10.1214/EJP.v15-762.Search in Google Scholar
[6] R. Garra, E. Orsingher, F. Polito, State-dependent fractional point processes. J. Appl. Probab. 52, No 1 (2015), 18–36; 10.1239/jap/1429282604.Search in Google Scholar
[7] R. Garra, E. Orsingher, M. Scavino, Some probabilistic properties of fractional point processes. Stoch. Anal. Appl. 35, No 4 (2017), 701–718; 10.1080/07362994.2017.1308831.Search in Google Scholar
[8] N. Gupta, A. Kumar, N. Leonenko, Tempered fractional Poisson processes and fractional equations with Z-transform. Stoch. Anal. Appl. 38, No 5 (2020), 939–957; 10.1080/07362994.2020.1748056.Search in Google Scholar
[9] K.K. Kataria, P. Vellaisamy, Saigo space-time fractional Poisson process via Adomian decomposition method. Statist. Probab. Lett. 129 (2017), 69-80; 10.1016/j.spl.2017.05.007.Search in Google Scholar
[10] K.K. Kataria, P. Vellaisamy, On distributions of certain state dependent fractional point processes. J. Theoret. Probab. 32, No 3 (2019), 1554–1580; 10.1007/s10959-018-0835-z.Search in Google Scholar
[11] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Search in Google Scholar
[12] N. Laskin, Fractional Poisson process. Commun. Nonlinear Sci. Numer. Simul. 8, No 3-4 (2003), 201–213; 10.1016/S1007-5704(03)00037-6.Search in Google Scholar
[13] M.M. Meerschaert, E. Nane, P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator. Electron. J. Probab. 16, No 59 (2011), 1600–1620; 10.1214/EJP.v16-920.Search in Google Scholar
[14] T.M. Michelitsch, A.P. Riascos, Generalized fractional Poisson process and related stochastic dynamics. Fract. Calc. Appl. Anal. 23, No 3 (2020), 656–693; 10.1515/fca-2020-0034.Search in Google Scholar
[15] E. Orsingher, F. Polito, Fractional pure birth processes. Bernoulli16, No 3 (2010), 858–881; 10.3150/09-BEJ235.Search in Google Scholar
[16] E. Orsingher, F. Polito, The space-fractional Poisson process. Statist. Probab. Lett. 82, No 4 (2012), 852–858; 10.1016/j.spl.2011.12.018.Search in Google Scholar
[17] E. Orsingher, C. Ricciuti, B. Toaldo, On semi-Markov processes and their Kolmogorov’s integro-differential equations. J. Funct. Anal. 275, No 4 (2018), 830–868; 10.1016/j.jfa.2018.02.011.Search in Google Scholar
[18] A. Pachon, F. Polito, C. Ricciuti, On discrete-time semi-Markov Processes. Discrete Contin. Dyn. Syst. Ser. B, (2020); 10.3934/dcdsb.2020170.Search in Google Scholar
[19] F. Polito, E. Scalas, A generalization of the space-fractional Poisson process and its connection to some Lévy processes. Electron. Commun. Probab. 21 (2016), 1–14; 10.1214/16-ECP4383.Search in Google Scholar
[20] C. Ricciuti, B. Toaldo, Semi-Markov models and motion in heterogeneous media. J. Stat. Phys. 169, No 2 (2017), 340–361; 10.1007/s10955-017-1871-2.Search in Google Scholar
[21] P. Vellaisamy, F. Veins, A probabilistic approach to Adomian polynomials. Stoch. Anal. Appl. (2020); 10.1080/07362994.2020.1755312.Search in Google Scholar
© 2020 Diogenes Co., Sofia
Articles in the same Issue
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–Volume 23–5–2020)
- Research Paper
- Tempered relaxation equation and related generalized stable processes
- Integrability properties of integral transforms via morrey spaces
- Fractional integro-differential equations in Wiener spaces
- Fractional fractals
- Two high-order time discretization schemes for subdiffusion problems with nonsmooth data
- Applications of Erdélyi-Kober fractional integral for solving time-fractional Tricomi-Keldysh type equation
- Analysis of fractional integro-differential equations with nonlocal Erdélyi-Kober type integral boundary conditions
- Regularity results for nonlocal evolution Venttsel’ problems
- Multivariate fractional phase–type distributions
- Trace inequalities for fractional integrals in mixed norm grand lebesgue spaces
- The asymptotic behavior of solutions of discrete nonlinear fractional equations
- State dependent versions of the space-time fractional poisson process
- Approximate controllability for stochastic fractional hemivariational inequalities of degenerate type
- Experimental investigation of fractional order behavior in an oscillating disk
- Cauchy problem for general time fractional diffusion equation
Articles in the same Issue
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–Volume 23–5–2020)
- Research Paper
- Tempered relaxation equation and related generalized stable processes
- Integrability properties of integral transforms via morrey spaces
- Fractional integro-differential equations in Wiener spaces
- Fractional fractals
- Two high-order time discretization schemes for subdiffusion problems with nonsmooth data
- Applications of Erdélyi-Kober fractional integral for solving time-fractional Tricomi-Keldysh type equation
- Analysis of fractional integro-differential equations with nonlocal Erdélyi-Kober type integral boundary conditions
- Regularity results for nonlocal evolution Venttsel’ problems
- Multivariate fractional phase–type distributions
- Trace inequalities for fractional integrals in mixed norm grand lebesgue spaces
- The asymptotic behavior of solutions of discrete nonlinear fractional equations
- State dependent versions of the space-time fractional poisson process
- Approximate controllability for stochastic fractional hemivariational inequalities of degenerate type
- Experimental investigation of fractional order behavior in an oscillating disk
- Cauchy problem for general time fractional diffusion equation
