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Power variation of multiple fractional integrals

  • Research Article
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Central European Journal of Mathematics

Abstract

We study the convergence in probability of the normalized q-variation of the multiple fractional multiparameter integral processes

$$\begin{gathered} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r = (t_1 ,...,t_r ) \to I_r^H (f_r )_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r } : = \int_{[0,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r ]} {f_r (s_1 ,...,s_r )dB_{s_1 }^H ...dB_{s_r }^H } , \hfill \\ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r = (t_1 ,...,t_r ) \to I_r^{H, - } (f_r )_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r } : = \int_{[0,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r ]} {f_r (s_1 ,...,s_r )dS_{s_1 }^H ...dS_{s_r }^H } , \hfill \\ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _2 = (t_1 ,t_2 ) \to I_r^H (g)_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _2 } : = \int_{[0,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _2 ]} {g(s_1 ,s_2 )dB_{s_1 }^{H,1} dB_{s_2 }^{H,2} } , \hfill \\ \end{gathered} $$

where f r, g are continuous deterministic functions, B H (resp. S H) is a fractional (resp. a sub-fractional) Brownian motion with Hurst parameter H > 1/2 and B H,1, B H,1 are independent fractional Brownian motions with Hurst parameter H.

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Authors and Affiliations

  1. Faculty of Mathematics and Informatics, University of Bucharest, Bucharest, Romania

    Constantin Tudor

  2. Department of Mathematics, Academy of Economical Studies, Bucharest, Romania

    Maria Tudor

Authors
  1. Constantin Tudor
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  2. Maria Tudor
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Dedicated to Professors Ioan Cuculescu and Marius Iosifescu on the occasion of their 70th birthdays.

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Tudor, C., Tudor, M. Power variation of multiple fractional integrals. centr.eur.j.math. 5, 358–372 (2007). https://doi.org/10.2478/s11533-007-0001-9

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  • DOI: https://doi.org/10.2478/s11533-007-0001-9

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