Properties of integrals with respect to fractional Poisson processes with compact kernels
Authors:
Y. Mishura and V. Zubchenko
Journal:
Theor. Probability and Math. Statist. 89 (2014), 143-152
MSC (2010):
Primary 60G22; Secondary 60G51
DOI:
https://doi.org/10.1090/S0094-9000-2015-00941-8
Published electronically:
January 26, 2015
MathSciNet review:
3235181
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Properties of a fractional Poisson process with the Molchan–Golosov kernel are studied. The kernel can be viewed as compact since it is non-zero on a compact interval. The integral of a nonrandom function with respect to centered and non-centered fractional Poisson processes with the Molchan–Golosov kernel is introduced. The second moments of these integrals are obtained in terms of the norm of the integrand in the space $L_{1/H}([0,T])$. Moment estimates for higher moments of these integrals are established by using the Bichteler–Jacod inequality.
References
- L. Beghin and E. Orsingher, Fractional Poisson process and related planar random motions, Bernoulli 12 (2006), no. 6, 1099–1126.
- Albert Benassi, Serge Cohen, and Jacques Istas, Identification and properties of real harmonizable fractional Lévy motions, Bernoulli 8 (2002), no. 1, 97–115. MR 1884160
- Céline Jost, Transformation formulas for fractional Brownian motion, Stochastic Process. Appl. 116 (2006), no. 10, 1341–1357. MR 2260738, DOI https://doi.org/10.1016/j.spa.2006.02.006
- Francesco Mainardi, Rudolf Gorenflo, and Enrico Scalas, A fractional generalization of the Poisson processes, Vietnam J. Math. 32 (2004), no. Special Issue, 53–64. MR 2120631
- Francesco Mainardi, Rudolf Gorenflo, and Alessandro Vivoli, Renewal processes of Mittag-Leffler and Wright type, Fract. Calc. Appl. Anal. 8 (2005), no. 1, 7–38. MR 2179226
- Carlo Marinelli, Claudia Prévôt, and Michael Röckner, Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise, J. Funct. Anal. 258 (2010), no. 2, 616–649. MR 2557949, DOI https://doi.org/10.1016/j.jfa.2009.04.015
- Tina Marquardt, Fractional Lévy processes with an application to long memory moving average processes, Bernoulli 12 (2006), no. 6, 1099–1126. MR 2274856, DOI https://doi.org/10.3150/bj/1165269152
- Mark M. Meerschaert, Erkan Nane, and P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator, Electron. J. Probab. 16 (2011), no. 59, 1600–1620. MR 2835248, DOI https://doi.org/10.1214/EJP.v16-920
- Jean Mémin, Yulia Mishura, and Esko Valkeila, Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion, Statist. Probab. Lett. 51 (2001), no. 2, 197–206. MR 1822771, DOI https://doi.org/10.1016/S0167-7152%2800%2900157-7
- Yuliya S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008. MR 2378138
- G. Molchan and J. Golosov, Gaussian stationary processes with asymptotic power spectrum, Soviet Mathematics Doklady 10 (1969), no. 1, 134–137.
- Vladas Pipiras and Murad S. Taqqu, Are classes of deterministic integrands for fractional Brownian motion on an interval complete?, Bernoulli 7 (2001), no. 6, 873–897. MR 1873833, DOI https://doi.org/10.2307/3318624
- Balram S. Rajput and Jan Rosiński, Spectral representations of infinitely divisible processes, Probab. Theory Related Fields 82 (1989), no. 3, 451–487. MR 1001524, DOI https://doi.org/10.1007/BF00339998
- O. N. Repin and A. I. Saichev, Fractional Poisson law, Radiophys. and Quantum Electronics 43 (2000), no. 9, 738–741 (2001). MR 1910034, DOI https://doi.org/10.1023/A%3A1004890226863
- Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
- Heikki Tikanmäki and Yuliya Mishura, Fractional Lévy processes as a result of compact interval integral transformation, Stoch. Anal. Appl. 29 (2011), no. 6, 1081–1101. MR 2847337, DOI https://doi.org/10.1080/07362994.2011.610172
- V. P. Zubchenko, Properties of solutions of stochastic differential equations with random coefficients, non-Lipschitz diffusion, and Poisson measures, Teor. Ĭmovīr. Mat. Stat. 82 (2010), 30–42 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 82 (2011), 11–26. MR 2790480, DOI https://doi.org/10.1090/S0094-9000-2011-00824-1
References
- L. Beghin and E. Orsingher, Fractional Poisson process and related planar random motions, Bernoulli 12 (2006), no. 6, 1099–1126.
- A. Benassi, S. Cohen, and J. Istas, Identification and properties of real harmonizable fractional Lévy motions, Bernoulli 8 (2002), 97–115. MR 1884160 (2003b:60075)
- C. Jost, Transformation formulas for fractional Brownian motion, Stoch. Process. Appl. 116 (2006), 1341–1357. MR 2260738 (2007e:60026)
- F. Mainardi, R. Gorenflo, and E. Scalas, A fractional generalization of the Poisson processes, Vietnam J. Math. 32 (2004), 53–64. MR 2120631
- F. Mainardi, R. Gorenflo, and A. Vivoli, Renewal processes of Mittag–Leffler and Wright type, Fract. Calc. Appl. Anal. 8 (2005), no. 1, 7–38. MR 2179226
- C. Marinelli, C. Prévôt, and M. Röckner, Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise, J. Funct. Anal. 258 (2010), no. 2, 616–649. MR 2557949 (2011a:60230)
- T. Marquardt, Fractional Lévy process with an application to long memory moving average processes, Bernoulli 12 (2006), no. 6, 1099–1126. MR 2274856 (2008j:60119)
- 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 (2012k:60252)
- J. Memin, Yu. Mishura, and E. Valkeila, Inequalities for the moments of Wiener integrals with respect to fractional Brownian motions, Statist. Probab. Lett. 51 (2001), 197–206. MR 1822771 (2002b:60096)
- Yu. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, Berlin, 2008. MR 2378138 (2008m:60064)
- G. Molchan and J. Golosov, Gaussian stationary processes with asymptotic power spectrum, Soviet Mathematics Doklady 10 (1969), no. 1, 134–137.
- V. Pipiras and M. Taqqu, Are classes of deterministic integrands for fractional Brownian motion on an interval complete?, Bernoulli 7 (2001), no. 6, 873–897. MR 1873833 (2002h:60176)
- B. S. Rajput and J. Rosinski, Spectral representaion of infinitely divisible processes, Probab. Theory Related Fields 82 (1989), 451–487. MR 1001524 (91i:60149)
- O. N. Repin and A. I. Saichev, Fractional Poisson law, Radiophysics and Quantum Electronics 43 (2000), no. 9, 738–741. MR 1910034
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, 1993. MR 1347689 (96d:26012)
- H. Tikanmaki and Yu. Mishura, Fractional Lévy process as a result of compact interval intergal transformation, Stoch. Anal. Appl. 29 (2011), 1081–1101. MR 2847337 (2012k:60116)
- V. P. Zubchenko, Properties of solutions of stochastic differential equations with random coefficients, non-Lipschitzian diffusion, and Poisson measures, Theor. Probab. Math. Statist. 82 (2011), 11–26. MR 2790480 (2012c:60161)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60G22,
60G51
Retrieve articles in all journals
with MSC (2010):
60G22,
60G51
Additional Information
Y. Mishura
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty for Mechanics and mathematics, Kyiv National Taras Shevchenko University, Volodymyrska St., 64, Kyiv, 01601, Ukraine
Email:
myus@univ.kiev.ua
V. Zubchenko
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Faculty for Mechanics and mathematics, Kyiv National Taras Shevchenko University, Volodymyrska St., 64, Kyiv, 01601, Ukraine
Email:
v_zubchenko@ukr.net
Keywords:
Fractional Poisson process,
integral representation of the fractional Poisson process,
Mandelbrot–van Ness kernel,
Molchan–Golosov kernel,
integral with respect to the fractional Poisson process,
Bichteler–Jacod inequality
Received by editor(s):
February 26, 2013
Published electronically:
January 26, 2015
Article copyright:
© Copyright 2015
American Mathematical Society