On the governing equations for Poisson and Skellam processes time-changed by inverse subordinators
Authors:
K. V. Buchak and L. M. Sakhno
Journal:
Theor. Probability and Math. Statist. 98 (2019), 91-104
MSC (2010):
Primary 60G50, 60G51, 60G55
DOI:
https://doi.org/10.1090/tpms/1064
Published electronically:
August 19, 2019
MathSciNet review:
3824680
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Additional Information
Abstract: In the paper we present the governing equations for marginal distributions of Poisson and Skellam processes time-changed by inverse subordinators. The equations are given in terms of convolution-type derivatives.
References
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References
- G. Aletti, N. N. Leonenko, and E. Merzbach, Fractional Poisson fields and martingales, J. Stat. Phys. 170 (2018), no. 4, 700–730. MR 3764004
- M. S. Alrawashdeh, J. F. Kelly, M. M. Meerschaert, and H.-P. Scheffler, Applications of inverse tempered stable subordinators, Comput. Math. Appl. 73 (2017), no. 6, 892–905. MR 3623093
- O. E. Barndorff-Nielsen, D. Pollard, and N. Shephard, Integer-valued Levy processes and low latency financial econometrics, Quant. Finance 12 (2011), no. 4, 587–605. MR 2909600
- L. Beghin and E. Orsingher, Fractional Poisson processes and related planar random motions, Electron J. Probab. 14 (2009), no. 61, 1790–1827. MR 2535014
- L. Beghin and E. Orsingher, Poisson-type processes governed by fractional and higher-order recursive differential equations, Electron J. Probab. 15 (2010), no. 22, 684–709. MR 2650778
- J. Bertoin, Lévy Processes, Cambridge University Press, Cambridge, 1996. MR 1406564
- K. Buchak and L. Sakhno, Compositions of Poisson and Gamma processes, Mod. Stoch. Theory Appl. 4 (2017), no. 2, 161–188. MR 3668780
- R. Garra, E. Orsingher, and M. Scavino, Some probabilistic properties of fractional point processes, Stoch. Anal. Appl. 35 (2017), no. 4, 701–718. MR 3651139
- A. Kerss, N. Leonenko, and A. Sikorskii, Fractional Skellam processes with applications to finance, Fract. Calc. Appl. Anal. 17 (2014), no. 2, 532–551. MR 3181070
- K. Kobylych and L. Sakhno, Point processes subordinated to compound Poisson processes, Theory Probab. Math. Statist. 94 (2017), 89–96.
- A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integral Equations Operator Theory 71 (2011), no. 4, 583–600. MR 2854867
- A. Kumar, E. Nane, and P. Vellaisamy, Time-changed Poisson processes, Statist. Probab. Lett. 81 (2011), 1899–1910. MR 2845907
- N. Leonenko, E. Scalas, and M. Trinh, The fractional non-homogeneous Poisson process. Statist. Probab. Lett. 120 (2017), 147–156. MR 3567934
- M. M. Meerschaert and H.-P. Scheffler, Triangular array limits for continuous time random walks, Stoch. Proc. Appl. 118 (2008), 1606–1633; 120 (2010), 2520–2521. MR 2728176
- M. M. Meerschaert, E. Nane, and P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator, Electron. J. Probab. 16 (2011), no. 59, 1600–1620. MR 2835248
- E. Orsingher and F. Polito, The space-fractional Poisson process, Statist. Probab. Lett. 82 (2012), 852–858. MR 2899530
- E. Orsingher and B. Toaldo, Counting processes with Bernstein intertimes and random jumps, J. Appl. Probab. 52 (2015), 1028–1044. MR 3439170
- K. Sato, Lévy processes and infinitely divisible distributions, Cambridge University Press, 1999. MR 1739520
- J. G. Skellam, The frequency distribution of the difference between two Poisson variables belonging to different populations, J. R. Stat. Soc. Ser. A (1946), 109–296. MR 0020750
- B. Toaldo, Convolution-type derivatives, hitting-times of subordinators and time-changed $C_0$-semigroups, Potential Anal. 42 (2015), 115–140. MR 3297989
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Additional Information
K. V. Buchak
Affiliation:
Mechanics and Mathematics Faculty, Taras Shevchenko National University of Kyiv, Volodymyrska 64, 01601, Kyiv, Ukraine
Email:
kristina.kobilich@gmail.com
L. M. Sakhno
Affiliation:
Mechanics and Mathematics Faculty, Taras Shevchenko National University of Kyiv, Volodymyrska 64, 01601, Kyiv, Ukraine
Email:
lms@univ.kiev.ua
Keywords:
Poisson process,
Skellam process,
time-change,
inverse subordinator,
governing equation,
convolution-type derivatives
Received by editor(s):
January 25, 2018
Published electronically:
August 19, 2019
Article copyright:
© Copyright 2019
American Mathematical Society
