Power variation of multiple fractional integrals

• Published in: Central European journal of mathematics 2007-06, Vol.5 (2), p.358-372
• Main Authors: Tudor, Constantin, Tudor, Maria
• Format: Article
• Language: English
• Subjects: 60H05; Brownian motion; Continuity (mathematics); Fractional Brownian motion; Fractional calculus; Integrals; multiple fractional Stratonovich integral; multiple fractional Wiener-Itô integral; Parameters; q-variation; sub-fractional Brownian motion
• ISSN: 1895-1074; 2391-5455; 1644-3616
• DOI: 10.2478/s11533-007-0001-9

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.