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Compositions of Poisson and Gamma processes
Volume 4, Issue 2 (2017), pp. 161–188
Khrystyna Buchak   Lyudmyla Sakhno  

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https://doi.org/10.15559/17-VMSTA79
Pub. online: 29 June 2017      Type: Research Article      Open accessOpen Access

Received
9 February 2017
Revised
16 May 2017
Accepted
2 June 2017
Published
29 June 2017

Abstract

In the paper we study the models of time-changed Poisson and Skellam-type processes, where the role of time is played by compound Poisson-Gamma subordinators and their inverse (or first passage time) processes. We obtain explicitly the probability distributions of considered time-changed processes and discuss their properties.

References

[1] 
Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press (2009). MR2512800. doi:10.1017/CBO9780511809781
[2] 
Barndorff-Nielsen, O.E., Pollard, D., Shephard, N.: Integer-valued Lévy processes and low latency financial econometrics. Quant. Finance 12(4), 587–605 (2011). MR2909600. doi:10.1080/14697688.2012.664935
[3] 
Bertoin, J.: Lévy Processes. Cambridge University Press (1996). MR1406564
[4] 
Cox, D.R.: Renewal Theory. Mathuen, London (1962). MR0153061
[5] 
Crescenzo, A.D., Martinucci, B., Zacks, S.: Compound Poisson process with a Poisson subordinator. J. Appl. Probab. 52(2), 360–374 (2015). MR3372080. doi:10.1239/jap/1437658603
[6] 
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2. Wiley (1971). MR0038583
[7] 
Garra, R., Orsingher, E., Scavino, M.: Some probabilistic properties of fractional point processes [Ona sechas opublikovana on line i ee koordinati slediushi]. Stoch. Anal. Appl. 35(4), 701–718 (2017). MR3651139. doi:10.1080/07362994.2017.1308831
[8] 
Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag-Leffler functions and their applications. J. Appl. Math. (2011), 51 pp. MR2800586. doi:10.1155/2011/298628
[9] 
Irwin, J.O.: The frequency distribution of the difference between two independent variates following the same Poisson distribution. J. R. Stat. Soc. A 100, 415–416 (1937).
[10] 
Kerss, A., Leonenko, N., Sikorskii, A.: Fractional Skellam processes with applications to finance. Fract. Calc. Appl. Anal. 17(2), 532–551 (2014). MR3181070. doi:10.2478/s13540-014-0184-2
[11] 
Kobylych, K., Sakhno, L.: Point processes subordinated to compound Poisson processes. Theory Probab. Math. Stat. 94, 85–92 (2016) (in Ukrainian); English translation to appear in Theory Probab. Math. Stat. 94 (2017). MR3553456
[12] 
Kumar, A., Nane, E., Vellaisamy, P.: Time-changed Poisson processes. Stat. Probab. Lett. 81(12), 1899–1910 (2011). MR2845907. doi:10.1016/j.spl.2011.08.002
[13] 
Leonenko, N., Scalas, E., Trinh, M.: The fractional non-homogeneous Poisson process. Stat. Probab. Lett. 120, 147–156 (2017). MR3567934. doi:10.1016/j.spl.2016.09.024
[14] 
Leonenko, N., Meerschaert, M., Schilling, R., Sikorskii, A.: Correlation Structure of Time-Changed Lévy Processes. Commun. Appl. Ind. Math. 6(1) (2014). MR3277310. doi:10.1685/journal.caim.483
[15] 
Orsingher, E., Polito, F.: The space-fractional Poisson process. Stat. Probab. Lett. 82, 852–858 (2012). MR2899530. doi:10.1016/j.spl.2011.12.018
[16] 
Orsingher, E., Toaldo, B.: Counting processes with Bernštein intertimes and random jumps. J. Appl. Probab. 52, 1028–1044 (2015). MR3439170. doi:10.1239/jap/1450802751
[17] 
Paris, R., Vinogradov, V.: Fluctuation properties of compound Poisson–Erlang Lévy processes. Commun. Stoch. Anal. 7(2), 283–302 (2013). MR3092235
[18] 
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press (1999). MR1739520
[19] 
Skellam, J.G.: The frequency distribution of the difference between two Poisson variables belonging to different populations. J. R. Stat. Soc. A, 109–296 (1946). MR0020750
[20] 
Sneddon, I.N.: Special Functions of Mathematical Physics and Chemistry. Oliver and Boyd, Edinburgh (1956). MR0080170
[21] 
Veillette, M., Taqqu, M.S.: Using differential equations to obtain joint moments of first-passage times of increasing Lévy processes. Stat. Probab. Lett. 80, 697–705 (2010). MR2595149. doi:10.1016/j.spl.2010.01.002

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Keywords
Time-change Poisson process Skellam process compound Poisson-Gamma subordinator inverse subordinator

MSC2010
60G50 60G51 60G55

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