Local properties of a multifractional stable field
Author:
Georgiy Shevchenko
Translated by:
The author
Journal:
Theor. Probability and Math. Statist. 85 (2012), 159-168
MSC (2010):
Primary 60G52, 60G17; Secondary 60G22, 60G18
DOI:
https://doi.org/10.1090/S0094-9000-2013-00882-5
Published electronically:
January 14, 2013
MathSciNet review:
2933711
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: An anisotropic harmonizable multifractional stable field is defined. Its continuity is proved. Existence and square integrability of local time are established. It is proved that the local time is jointly continuous in the Gaussian case.
References
- Antoine Ayache, The generalized multifractional field: a nice tool for the study of the generalized multifractional Brownian motion, J. Fourier Anal. Appl. 8 (2002), no. 6, 581–601. MR 1932747, DOI https://doi.org/10.1007/s00041-002-0028-z
- Albert Benassi, Stéphane Jaffard, and Daniel Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoamericana 13 (1997), no. 1, 19–90 (English, with English and French summaries). MR 1462329, DOI https://doi.org/10.4171/RMI/217
- B. Boufoussi, M. Dozzi, and R. Guerbaz, On the local time of multifractional Brownian motion, Stochastics 78 (2006), no. 1, 33–49. MR 2219711, DOI https://doi.org/10.1080/17442500600578073
- Marco Dozzi and Georgiy Shevchenko, Real harmonizable multifractional stable process and its local properties, Stochastic Process. Appl. 121 (2011), no. 7, 1509–1523. MR 2802463, DOI https://doi.org/10.1016/j.spa.2011.03.012
- Donald Geman and Joseph Horowitz, Occupation densities, Ann. Probab. 8 (1980), no. 1, 1–67. MR 556414
- Erick Herbin, From $N$ parameter fractional Brownian motions to $N$ parameter multifractional Brownian motions, Rocky Mountain J. Math. 36 (2006), no. 4, 1249–1284. MR 2274895, DOI https://doi.org/10.1216/rmjm/1181069415
- Norio Kôno and Makoto Maejima, Self-similar stable processes with stationary increments, Stable processes and related topics (Ithaca, NY, 1990) Progr. Probab., vol. 25, Birkhäuser Boston, Boston, MA, 1991, pp. 275–295. MR 1119361
- S. C. Lim and L. P. Teo, Sample path properties of fractional Riesz-Bessel field of variable order, J. Math. Phys. 49 (2008), no. 1, 013509, 31. MR 2385268, DOI https://doi.org/10.1063/1.2830431
- M. A. Lifshits, Gaussian random functions, Mathematics and its Applications, vol. 322, Kluwer Academic Publishers, Dordrecht, 1995. MR 1472736
- M. B. Marcus and G. Pisier, Characterizations of almost surely continuous $p$-stable random Fourier series and strongly stationary processes, Acta Math. 152 (1984), no. 3-4, 245–301. MR 741056, DOI https://doi.org/10.1007/BF02392199
- Yuliya S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008. MR 2378138
- R. F. Peltier and J. Lévy Véhel, Multifractional Brownian motion: definition and preliminary results, INRIA Research Report, no. 2645, 1995.
- K. V. Ral′chenko and G. M. Shevchenko, Properties of the paths of a multifractal Brownian motion, Teor. Ĭmovīr. Mat. Stat. 80 (2009), 106–116 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 80 (2010), 119–130. MR 2541957, DOI https://doi.org/10.1090/S0094-9000-2010-00799-X
- Gennady Samorodnitsky and Murad S. Taqqu, Stable non-Gaussian random processes, Stochastic Modeling, Chapman & Hall, New York, 1994. Stochastic models with infinite variance. MR 1280932
- Yimin Xiao, Properties of local-nondeterminism of Gaussian and stable random fields and their applications, Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), no. 1, 157–193 (English, with English and French summaries). MR 2225751
- Yimin Xiao, Sample path properties of anisotropic Gaussian random fields, A minicourse on stochastic partial differential equations, Lecture Notes in Math., vol. 1962, Springer, Berlin, 2009, pp. 145–212. MR 2508776, DOI https://doi.org/10.1007/978-3-540-85994-9_5
References
- A. Ayache, The generalized multifractional field: A nice tool for the study of the generalized multifractional Brownian motion, J. Fourier Anal. Appl. 8 (2002), no. 6, 581–601. MR 1932747 (2003j:60050)
- A. Benassi, S. Jaffard, and D. Roux, Gaussian processes and pseudodifferential elliptic operators, Revista Mathematica Iberoamericana 13 (1997), no. 1, 19–89. MR 1462329 (98k:60056)
- B. Boufoussi, M. Dozzi, and R. Guerbaz, On the local time of multifractional Brownian motion, Stochastics 78 (2006), no. 1, 33–49. MR 2219711 (2007m:60239)
- M. Dozzi and G. Shevchenko, Real harmonizable multifractional stable process and its local properties, Stochastic Processes Appl. 121 (2011), no. 7, 1509–1523. MR 2802463 (2012g:60163)
- D. Geman and J. Horowitz, Occupation densities, Ann. Probab. 8 (1980), 1–67. MR 556414 (81b:60076)
- E. Herbin, From $N$ parameter fractional Brownian motions to $N$ parameter multifractional Brownian motions, Rocky Mountain J. Math. 36 (2006), no. 4, 1249–1284. MR 2274895 (2007m:62051)
- N. Kôno and M. Maejima, Self-similar stable processes with stationary increments, Stable Processes and Related Topics, Sel. Pap. Workshop, Ithaca/NY (USA), 1990, Progress in Probabability, vol. 25, 1991, pp. 275–295. MR 1119361 (92j:60052)
- S. C. Lim and L. P. Teo, Sample path properties of fractional Riesz-Bessel field of variable order, J. Math. Phys. 49 (2008), no. 1, 013509. MR 2385268 (2008k:60085)
- M. A. Lifshits, Gaussian Random Functions, Mathematics and its Applications (Dordrecht), 322, Kluwer Academic Publishers, Dordrecht, 1995. MR 1472736 (98k:60059)
- M. B. Marcus and G. Pisier, Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes, Acta Math. 152 (1984), 245–301. MR 741056 (86b:60069)
- Yu. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, Berlin, 2008. MR 2378138 (2008m:60064)
- R. F. Peltier and J. Lévy Véhel, Multifractional Brownian motion: definition and preliminary results, INRIA Research Report, no. 2645, 1995.
- K. V. Ralchenko and G. M. Shevchenko, Path properties of multifractal Brownian motion, Theor. Probability Math. Statist. 80 (2010), 119–130. MR 2541957 (2010h:60122)
- G. Samorodnitsky and M. S. Taqqu, Stable non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Stochastic Modeling, Chapman & Hall, New York, NY, 1994. MR 1280932 (95f:60024)
- Y. Xiao, Properties of local nondeterminism of Gaussian and stable random fields and their applications, Ann. Fac. Sci. Toulouse Math. XV (2006), 157–193. MR 2225751 (2008e:60152)
- Y. Xiao, Sample path properties of anisotropic Gaussian random fields, A Minicourse on Stochastic Partial Differential Equations (D. Khoshnevisan and F. Rassoul-Agha, eds.), Lecture Notes in Math., vol. 1962, Springer, New York, 2009, pp. 145–212. MR 2508776 (2010i:60159)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60G52,
60G17,
60G22,
60G18
Retrieve articles in all journals
with MSC (2010):
60G52,
60G17,
60G22,
60G18
Additional Information
Georgiy Shevchenko
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Mechanics and Mathematics Faculty, Kyiv National Taras Shevchenko University, 60 Volodymyrska, 01601 Kyiv, Ukraine
Email:
zhora@univ.kiev.ua
Keywords:
Stable process,
harmonizable process,
multifractionality,
local time,
local non-determinism
Received by editor(s):
August 20, 2029
Published electronically:
January 14, 2013
Additional Notes:
The author is grateful to the European commission for support of research within the “Marie Curie Actions” program, grant PIRSES-GA-2008-230804
Article copyright:
© Copyright 2013
American Mathematical Society
