Modern Stochastics: Theory and Applications logo


  • Help
Login Register

  1. Home
  2. Issues
  3. Volume 11, Issue 4 (2024)
  4. Sample path properties of multidimension ...

Modern Stochastics: Theory and Applications

Submit your article Information Become a Peer-reviewer
  • Article info
  • Full article
  • Related articles
  • More
    Article info Full article Related articles

Sample path properties of multidimensional integral with respect to stochastic measure
Volume 11, Issue 4 (2024), pp. 421–437
Boris Manikin ORCID icon link to view author Boris Manikin details   Vadym Radchenko  

Authors

 
Placeholder
https://doi.org/10.15559/24-VMSTA256
Pub. online: 30 May 2024      Type: Research Article      Open accessOpen Access

Received
11 January 2024
Revised
24 March 2024
Accepted
30 April 2024
Published
30 May 2024

Abstract

The integral with respect to a multidimensional stochastic measure, assuming only its σ-additivity in probability, is studied. The continuity and differentiability of realizations of the integral are established.

References

[1] 
Adler, R.J.: The Geometry of Random Fields. John Wiley & Sons, Chichester (1981). MR0611857
[2] 
Andrianov, P., Skopina, M.: On Jackson-type inequalities associated with separable Haar wavelets. Int. J. Wavelets Multiresolut. Inf. Process. 14(03), 1650005 (2016). MR3503421. https://doi.org/10.1142/S0219691316500053
[3] 
Bai, S.: Limit theorems for conservative flows on multiple stochastic integrals. J. Theor. Probab. 35, 917–948 (2022). MR4414409. https://doi.org/10.1007/s10959-021-01090-9
[4] 
Besov, O.V., Il’in, V.P., Nikol’skii, S.M.: Integral Representations of Functions and Imbedding Theorems, V. 1, 2. John Wiley & Sons, New York (1978, 1979). MR0521808
[5] 
Biermé, H., Lacaux, C.: Hölder regularity for operator scaling stable random fields. Stochastic Process. Appl. 119(7), 2222–2248 (2009). MR2531090. https://doi.org/10.1016/j.spa.2008.10.008
[6] 
Biermé, H., Lacaux, C.: Modulus of continuity of some conditionally sub-Gaussian fields, application to stable random fields. Bernoulli 21(3), 1719–1759 (2015). MR3352059. https://doi.org/10.3150/14-BEJ619
[7] 
Clarke De la Cerda, J., Tudor, C.A.: Wiener integrals with respect to the Hermite random field and applications to the wave equation. Collect. Math. 65(3), 341–356 (2014). MR3240998. https://doi.org/10.1007/s13348-014-0108-9
[8] 
DeVore, R.A., Popov, V.A.: Interpolation of Besov spaces. Trans. Amer. Math. Soc. 305(1), 397–414 (1988). MR0920166. https://doi.org/10.2307/2001060
[9] 
Gikhman, I.I., Skorokhod, A.V.: Introduction to the Theory of Random Processes. Dover Publications Inc, New York (1996). MR1435501
[10] 
Harang, F.A.: An extension of the sewing lemma to hyper-cubes and hyperbolic equations driven by multi-parameter Young fields. Stoch. Partial Differ. Equ. Anal. Comput. 9(1), 746–788 (2021). MR4297239. https://doi.org/10.1007/s40072-020-00184-5
[11] 
Hinojosa-Calleja, A.: Exact uniform modulus of continuity for q-isotropic Gaussian random fields. Statist. Probab. Lett. 197, 109813 (2023). MR4557375. https://doi.org/10.1016/j.spl.2023.109813
[12] 
Kamont, A.: A discrete characterization of Besov spaces. Approx. Theory Appl. 13, 63–77 (1997). MR1750304
[13] 
Kashin, B.S., Saakyan, A.A.: Orthogonal Series vol. 75. American Mathematical Soc. (1989). MR1007141. https://doi.org/10.1090/mmono/075
[14] 
Krylov, N.V.: Introduction to the Theory of Random Processes vol. 43. American Mathematical Soc., Providence (2002). MR1885884. https://doi.org/10.1090/gsm/043
[15] 
Kwapień, S., Woyczyński, W.A.: Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston (1992). MR1167198. https://doi.org/10.1007/978-1-4612-0425-1
[16] 
Leoni, G.: A First Course in Sobolev Spaces. AMS, Providence (2017). MR3726909. https://doi.org/10.1090/gsm/181
[17] 
Nualart, D.: Malliavin Calculus and Related Topic, 2nd edn. Springer (2006) MR2200233
[18] 
Panigrahia, S., Royb, P., Xiao, Y.: Maximal moments and uniform modulus of continuity for stable random fields. Stochastic Process. Appl. 136, 92–124 (2021). MR4238104. https://doi.org/10.1016/j.spa.2021.02.002
[19] 
Radchenko, V.: Sample functions of stochastic measures and Besov spaces. Theory Probab. Appl. 54, 160–168 (2010). MR2766653. https://doi.org/10.1137/S0040585X97984048
[20] 
Radchenko, V.: General Stochastic Measures: Integration, Path Properties, and Equations. Wiley – ISTE, London (2022) MR4687103
[21] 
Radchenko, V.M., Stefans’ka, N.O.: Fourier series and Fourier-Haar series for stochastic measures. Theory Probab. Math. Statist. 99(2), 203–211 (2018) MR3666879. https://doi.org/10.1090/tpms/1041
[22] 
Romanyuk, V.S.: Multiple Haar basis and its properties. Ukrainian Math. J. 67(9), 1411–1425 (2016). MR3473729. https://doi.org/10.1007/s11253-016-1162-0
[23] 
Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations. Walter de Gruyter, Berlin (1996). MR1419319. https://doi.org/10.1515/9783110812411
[24] 
Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman and Hall, London (1994). MR1280932
[25] 
Sottinen, T., Tudor, C.A.: On the equivalence of multiparameter Gaussian processes. J. Theor. Probab. 19(2), 461–485 (2006). MR2283386. https://doi.org/10.1007/s10959-006-0022-5
[26] 
Sottinen, T., Viitasaari, L.: Fredholm representation of multiparameter Gaussian processes with applications to equivalence in law and series expansions. Mod. Stoch. Theory Appl. 2(3), 287–295 (2015). MR3407507. https://doi.org/10.15559/15-VMSTA39CNF
[27] 
Triebel, H.: Theory of Function Spaces. Geest & Portig K.-G. and Birkhäuser, Leipzig and Basel (1983). MR0730762. https://doi.org/10.1007/978-3-0346-0416-1
[28] 
Triebel, H.: The Structure of Functions. Birkhäuser, Basel (2001). MR3013187
[29] 
Tudor, C.: Non-Gaussian Selfsimilar Stochastic Processes. Springer (2023). MR4647498. https://doi.org/10.1007/978-3-031-33772-7
[30] 
Xiao, Y.: In: Khoshnevisan, D., Rassoul-Agha, F. (eds.) Sample Path Properties of Anisotropic Gaussian Random Fields, pp. 145–212. Springer, Berlin, Heidelberg (2009). MR2508776. https://doi.org/10.1007/978-3-540-85994-9_5
[31] 
Xiao, Y.: Uniform modulus of continuity of random fields. Monatsh. Math. 159, 163–184 (2010). MR2564392. https://doi.org/10.1007/s00605-009-0133-z

Full article Related articles PDF XML
Full article Related articles PDF XML

Copyright
© 2024 The Author(s). Published by VTeX
by logo by logo
Open access article under the CC BY license.

Keywords
Stochastic measure trajectories of random functions multivariable Haar functions Besov space

MSC2010
60H05 60G60 60G17

Metrics
since March 2018
574

Article info
views

127

Full article
views

171

PDF
downloads

66

XML
downloads

Export citation

Copy and paste formatted citation
Placeholder

Download citation in file


Share


RSS

MSTA

MSTA

  • Online ISSN: 2351-6054
  • Print ISSN: 2351-6046
  • Copyright © 2018 VTeX

About

  • About journal
  • Indexed in
  • Editors-in-Chief

For contributors

  • Submit
  • OA Policy
  • Become a Peer-reviewer

Contact us

  • ejournals-vmsta@vtex.lt
  • Mokslininkų 2A
  • LT-08412 Vilnius
  • Lithuania
Powered by PubliMill  •  Privacy policy