Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


IMPACT FACTOR 2018: 3.514
5-year IMPACT FACTOR: 3.524

CiteScore 2018: 3.44

SCImago Journal Rank (SJR) 2018: 1.891
Source Normalized Impact per Paper (SNIP) 2018: 1.808

Mathematical Citation Quotient (MCQ) 2017: 0.98

Online
ISSN
1314-2224
See all formats and pricing
More options …
Volume 17, Issue 2

Issues

Fractional Skellam processes with applications to finance

Alexander Kerss / Nikolai Leonenko / Alla Sikorskii
  • Department of Statistics and Probability, Michigan State University, 619 Red Cedar Road, East Lansing, MI, 48824, USA
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2014-03-21 | DOI: https://doi.org/10.2478/s13540-014-0184-2

Abstract

The recent literature on high frequency financial data includes models that use the difference of two Poisson processes, and incorporate a Skellam distribution for forward prices. The exponential distribution of inter-arrival times in these models is not always supported by data. Fractional generalization of Poisson process, or fractional Poisson process, overcomes this limitation and has Mittag-Leffler distribution of inter-arrival times. This paper defines fractional Skellam processes via the time changes in Poisson and Skellam processes by an inverse of a standard stable subordinator. An application to high frequency financial data set is provided to illustrate the advantages of models based on fractional Skellam processes.

MSC: Primary 60E05, 60G22; Secondary 60G51, 26A33

Keywords: fractional Poisson process; fractional Skellam process; Mittag-Leffler distribution; high frequency financial data

  • [1] E. Bacry, M. Delattre, M. Hoffman and J. Muzy, Modeling microstructure noise with mutually exciting point processes. Quant. Finance 13 No 1 (2013), 65–77. http://dx.doi.org/10.1080/14697688.2011.647054Web of ScienceCrossrefGoogle Scholar

  • [2] E. Bacry, M. Delattre, M. Hoffman and J. Muzy, Some limit theorems for Hawkes processes and applications to financial statistics. Stoch. Proc. Appl. 123, No 7 (2013), 2475–2499. http://dx.doi.org/10.1016/j.spa.2013.04.007Web of ScienceCrossrefGoogle Scholar

  • [3] O. E. Barndorff-Nielsen, D. Pollard and N. Shephard, Integer-valued Lévy processes and low latency financial econometrics. Quant. Finance 12, No 4 (2011), 587–605. Web of ScienceGoogle Scholar

  • [4] L. Beghin and E. Orsingher, Fractional Poisson processes and related random motions. Electron. J. Probab. 14 (2009), 1790–1826. http://dx.doi.org/10.1214/EJP.v14-675CrossrefGoogle Scholar

  • [5] L. Beghin and C. Macci, Large deviations for fractional Poisson process. Statistics and Probability Letters 83, No 4 (2013), 1193–1202. http://dx.doi.org/10.1016/j.spl.2013.01.017CrossrefWeb of ScienceGoogle Scholar

  • [6] L. Beghin and C. Macci, Fractional discrete processes: compound Poisson and mixed Poisson representations. Preprint available at arXiv:1303.2861v1 [math.PR] (2013). Google Scholar

  • [7] N.H. Bingham, Limit theorems for occupation times of Markov processes. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 (1971), 1–22. http://dx.doi.org/10.1007/BF00538470CrossrefGoogle Scholar

  • [8] L. Bondesson, G. Kristiansen, and F. Steutel, Infinite divisibility of random variables and their integer parts. Statistics and Probability Letters 28 (1996), 271–278. http://dx.doi.org/10.1016/0167-7152(95)00135-2CrossrefGoogle Scholar

  • [9] D. Cahoy and V. Uchaikin and A. Woyczynski, Parameter estimation from fractional Poisson process. J. Statist. Plann. Inference 140, No 11 (2013), 3106–3120. http://dx.doi.org/10.1016/j.jspi.2010.04.016CrossrefGoogle Scholar

  • [10] P. Carr, Semi-static hedging of barrier options under Poisson jumps. Int. J. Theor. Appl. Finance 14, No 7 (2011), 1091–1111. http://dx.doi.org/10.1142/S0219024911006668CrossrefGoogle Scholar

  • [11] R. Gorenflo, Yu. Luchko and F. Mainardi, Analytical properties and applications of the Wright function. Fract. Calc. Appl. Anal. 2, No 4 (1999), 383–414. Google Scholar

  • [12] M. G. Hahn, K. Kobayashi and S. Umarov, Fokker-Planck-Kolmogorov equaions associated with time-changed Brownian motion. Proc. Amer. Math. Soc. 139 (2011), 691–705. http://dx.doi.org/10.1090/S0002-9939-2010-10527-0Web of ScienceCrossrefGoogle Scholar

  • [13] H.J. Haubold, A.M. Mathai and R.K. Saxena, Mittag-Leffler functions and their applications. J. Appl. Math. 2011 (2011), Article ID 298628, 51 pages. Google Scholar

  • [14] C. C. Heyde and N. N. Leonenko, Student processes. Adv. Appl. Prob. 37 (2005), 342–365. http://dx.doi.org/10.1239/aap/1118858629CrossrefGoogle Scholar

  • [15] S. Howison and D. Lamper, Trading volume in models of financial derivatives. Applied Mathematical Finance 8 (2001), 119–135. http://dx.doi.org/10.1080/13504860110074163CrossrefGoogle Scholar

  • [16] J.O. Irwin, The frequency distribution of the difference between two independent variates following the same Poisson distribution. J. of the Royal Statistical Society, Ser. A, 100 (1937), 415–416. http://dx.doi.org/10.2307/2980526CrossrefGoogle Scholar

  • [17] N. Laskin, Fractional Poisson process. Commun. Nonlinear Sci. Numer. Simul., 8 (2003), 201–213. http://dx.doi.org/10.1016/S1007-5704(03)00037-6CrossrefGoogle Scholar

  • [18] N. N. Leonenko, S. Petherick and A. Sikorskii, Fractal activity time models for risky asset with dependence and generalized hyperbolic distributions. Stochastic Analysis and Applications 30, No 3 (2012), 476–492. http://dx.doi.org/10.1080/07362994.2012.668443Web of ScienceCrossrefGoogle Scholar

  • [19] N. N. Leonenko, M. M. Meerschaert, R. Schilling and A. Sikorskii, Correlation structure of time-changed Lévy processes. Preprint, available at http://www.stt.msu.edu/users/mcubed/CTRWcorrelation.pdf (2014). Google Scholar

  • [20] F. Mainardi and R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes. J. Computational and Applied Mathematics 118, No 1–2 (2000), 283–299. http://dx.doi.org/10.1016/S0377-0427(00)00294-6CrossrefGoogle Scholar

  • [21] F. Mainardi, R. Gorenflo and E. Scalas, A fractional generalization of the Poisson processes. Vietnam Journ. Math. 32 (2004), 53–64. Google Scholar

  • [22] F. Mainardi, R. Gorenflo, A. Vivoli, Beyond the Poisson renewal process: A tutorial survey. J. Comput. Appl. Math. 205 (2007), 725–735. http://dx.doi.org/10.1016/j.cam.2006.04.060CrossrefWeb of ScienceGoogle Scholar

  • [23] M. M. Meerschaert, E. Nane and P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator. Electronic J. of Probability 16 (2011), Paper No. 59, 1600–1620. http://dx.doi.org/10.1214/EJP.v16-920CrossrefGoogle Scholar

  • [24] M. M. Meerschaert, E. Nane and P. Vellaisamy, Fractional Cauchy problems on bounded domains. Annals of Probability 37, No 3 (2009), 979–1007. http://dx.doi.org/10.1214/08-AOP426Web of ScienceCrossrefGoogle Scholar

  • [25] M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus. De Gruyter, Berlin (2012). Google Scholar

  • [26] M. M. Meerschaert and H.-P. Scheffler, Triangular array limits for continuous time random walks. Stoch. Proc. Appl. 118 (2008), 1606–1633. http://dx.doi.org/10.1016/j.spa.2007.10.005CrossrefGoogle Scholar

  • [27] M. M. Meerschaert, R. Schilling and A. Sikorskii, Stochastic solutions for fractional wave equations. Nonlinear Dynamics (2014), To appear. Preprint available at: http://www.stt.msu.edu/users/mcubed/waveCTRW.pdf. Google Scholar

  • [28] O. N. Repin and A. I. Saichev, Fractional Poisson law. Radiophys. and Quantum Electronics 43 (2000), 738–741. http://dx.doi.org/10.1023/A:1004890226863CrossrefGoogle Scholar

  • [29] E. Scalas and N. Viles, On the convergence of quadratic variation for compound fractional Poisson process. Fract. Calc. Appl. Anal. 15, No 2 (2012), 314–331; DOI: 10.2478/s13540-012-0023-2; http://link.springer.com/article/10.2478/s13540-012-0023-2. CrossrefWeb of ScienceGoogle Scholar

  • [30] J. G. Skellam, The frequency distribution of the difference between two Poisson variables belonging to different populations. J. of the Royal Statistical Society, Ser. A (1946), 109–296. Google Scholar

  • [31] I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry. Oliver and Boyd, Edinburgh — London; Intersci. Publ., New York (1956); New Ed.: Longman, Harlow (1980). Google Scholar

  • [32] V. V. Uchaikin, D. O. Cahoy, R. T. Sibatov, Fractional processes: from Poisson to branching one. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 18 (2008), 2717–2725. http://dx.doi.org/10.1142/S0218127408021932CrossrefGoogle Scholar

  • [33] M. Veillette and M. S. Taqqu, Using differential equations to obtain joint moments of first-passage times of increasing Lévy processes. Statist. Probab. Lett. 80 (2010), 697–705. http://dx.doi.org/10.1016/j.spl.2010.01.002Web of ScienceCrossrefGoogle Scholar

About the article

Published Online: 2014-03-21

Published in Print: 2014-06-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 17, Issue 2, Pages 532–551, ISSN (Online) 1314-2224, DOI: https://doi.org/10.2478/s13540-014-0184-2.

Export Citation

© 2014 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Arun Kumar, Nikolai Leonenko, and Alois Pichler
Mathematics and Financial Economics, 2019
[2]
Jean-Philippe Aguilar, Cyril Coste, and Jan Korbel
Fractional Calculus and Applied Analysis, 2018, Volume 21, Number 4, Page 981
[3]
Neil Shephard and Justin J. Yang
Journal of the American Statistical Association, 2017, Volume 112, Number 519, Page 1090
[4]
Khrystyna Buchak and Lyudmyla Sakhno
Modern Stochastics: Theory and Applications, 2017, Volume 4, Number 2, Page 161
[5]
Valentina V. Tarasova and Vasily E. Tarasov
Communications in Nonlinear Science and Numerical Simulation, 2018, Volume 55, Page 127
[6]
Rong Lu, Ryan M Smith, Michal Seweryn, Danxin Wang, Katherine Hartmann, Amy Webb, Wolfgang Sadee, and Grzegorz A Rempala
BMC Genomics, 2015, Volume 16, Number 1

Comments (0)

Please log in or register to comment.
Log in