Properties of Poisson processes directed by compound Poisson-Gamma subordinators
Volume 5, Issue 2 (2018), pp. 167–189
Pub. online: 2 May 2018
Type: Research Article
Open Access
Open Access
Received
12 January 2018
12 January 2018
Revised
7 April 2018
7 April 2018
Accepted
11 April 2018
11 April 2018
Published
2 May 2018
2 May 2018
Abstract
In the paper we consider time-changed Poisson processes where the time is expressed by compound Poisson-Gamma subordinators $G(N(t))$ and derive the expressions for their hitting times. We also study the time-changed Poisson processes where the role of time is played by the processes of the form $G(N(t)+at)$ and by the iteration of such processes.
References
Aletti, G., Leonenko, N.N., Merzbach, E.: Fractional Poisson fields and martingales. Journal of Statistical Physics MR3764004. https://doi.org/10.1007/s10955-018-1951-y
Applebaum, D.: Lévy Processes and Stochastic Calculus (second edition). Cambridge University Press (2009) MR2512800. https://doi.org/10.1017/CBO9780511809781
Beghin, L., D’Ovidio, M.: Fractional Poisson process with random drift. Electron. J. Probab. 19 (2014) MR3304182. https://doi.org/10.1214/EJP.v19-3258
Beghin, L., Orsingher, E.: Population processes sampled at random times. J. Stat. Phys. 163, 1–21 (2016) MR3472091. https://doi.org/10.1007/s10955-016-1475-2
Buchak, K., Sakhno, L.: Compositions of Poisson and Gamma processes. Mod. Stoch.: Theory Appl. 4(2), 161–188 (2017) MR3668780. https://doi.org/10.15559/17-VMSTA79
Crescenzo, A.D., Martinucci, B., Zacks, S.: Compound Poisson process with a Poisson subordinator. J. Appl. Probab. 52(2), 360–374 (2015) MR3372080. https://doi.org/10.1239/jap/1437658603
Garra, R., Orsingher, E., Scavino, M.: Some probabilistic properties of fractional point processes. Stoch. Anal. Appl. 35(4), 701–718 (2017) MR3651139. https://doi.org/10.1080/07362994.2017.1308831
Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag-Leffler functions and their applications. Journal of Applied Mathematics 51 (2011) MR2800586. https://doi.org/10.1155/2011/298628
Kobylych, K., Sakhno, L.: Point processes subordinated to compound Poisson processes. Theory Probab. Math. Stat. 94, 85–92 (2016). (in Ukrainian); English translation in Theory of Probability and Mathematical Statistics 94, 89–96 (2017) MR3553456. https://doi.org/10.1090/tpms/1011
Leonenko, N., Scalas, E., Trinh, M.: The fractional non-homogeneous Poisson process. Stat. Probab. Lett. 120, 147–156 (2017) MR3567934. https://doi.org/10.1016/j.spl.2016.09.024
Orsingher, E., Polito, F.: Compositions, random sums and continued random fractions of Poisson and fractional Poisson processes. J. Stat. Phys. 148, 233–249 (2012) MR2966360. https://doi.org/10.1007/s10955-012-0534-6
Orsingher, E., Polito, F.: The space-fractional Poisson process. Stat. Probab. Lett. 82, 852–858 (2012) MR2899530. https://doi.org/10.1016/j.spl.2011.12.018
Orsingher, E., Polito, F.: On the integral of fractional Poisson processes. Stat. Probab. Lett. 83(4), 1006–1017 (2013) MR3041370. https://doi.org/10.1016/j.spl.2012.12.016
Orsingher, E., Toaldo, B.: Counting processes with Bernštein intertimes and random jumps. J. Appl. Probab. 52, 1028–1044 (2015) MR3439170. https://doi.org/10.1239/jap/1450802751
Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press (1999) MR1739520
Sneddon, I.N.: Special functions of mathematical physics and chemistry. Oliver and Boyd, Edinburgh (1956) MR0080170
Watanabe, T.: On Bessel transforms of multimodal increasing Lévy processes. Jpn. J. Math. 25(2), 227–256 (1999) MR1735462. https://doi.org/10.4099/math1924.25.227
