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2007 | 5 | 2 | 358-372
Tytuł artykułu

Power variation of multiple fractional integrals

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
5
Numer
2
Strony
358-372
Opis fizyczny
Daty
wydano
2007-06-01
online
2007-01-26
Twórcy
autor
  • Academy of Economical Studies
Bibliografia
  • [1] E. Alòs and D. Nualart: “Stochastic integration with respect to the fractional Brownian motion”, Stoch. Stoch. Rep., Vol. 75, (2003), pp. 277–305.
  • [2] X. Bardina, M. Jolis and C.A. Tudor: “Weak approximation of the multiple integrals with respect to the fractional Brownian motion”, Stoch. Proc. Appl., Vol. 105, (2003), pp. 315–344. http://dx.doi.org/10.1016/S0304-4149(03)00018-8
  • [3] O.E. Barndorff-Nielsen and N. Shephard: “Econometric Analysis of realized volatility and its use in estimating stochastic volatility models”, J. Roy. Stat. Soc., B, Vol. 64, (2002), pp. 255–280.
  • [4] O.E. Barndorff-Nielsen and N. Shephard: “Realized power variation and stochastic volatility models”, Bernoulli, Vol. 9, (2003), pp. 243–265.
  • [5] O.E. Barndorff-Nielsen and N. Shephard: “Power and bipower with stochastic volatility and jumps” (with discussion), J. Financial Econometrics, Vol. 2, (2004), pp. 1–48. http://dx.doi.org/10.1093/jjfinec/nbh001
  • [6] O.E. Barndorff-Nielsen and N. Shephard: “Econometric Analysis of realized covariation: high frequency covariance, regression and correlation in financial economics”, Econometrica, Vol. 72, (2004), pp. 885–925. http://dx.doi.org/10.1111/j.1468-0262.2004.00515.x
  • [7] T. Bojdecki, L. Gorostiza and A. Talarczyk: “Sub-fractional Brownian motion and its relation to occupation times”, Stat. & Probab. Lett., Vol. 69, (2004), pp. 405–419. http://dx.doi.org/10.1016/j.spl.2004.06.035
  • [8] P. Caithamer: “Decoupled double stochastic fractional integrals”, Stochastics, Vol. 77, Vol. 3, (2005), pp. 205–210.
  • [9] J.M. Corcuera, D. Nualart and J.C. Woerner: “Power variation of some integral long-memory processes”, Bernoulli, Vol. 14(4), (2006), pp. 713–735. http://dx.doi.org/10.3150/bj/1155735933
  • [10] A. Dasgupta and G. Kallianpur: “Chaos decomposition of multiple fractional integrals and applications”, Probab. Th. Rel. Fields, Vol. 115, (1999), pp. 505–525. http://dx.doi.org/10.1007/s004400050247
  • [11] A. Dasgupta and G. Kallianpur: “Multiple fractional integrals”, Probab. Th. Rel. Fields, Vol. 115, (1999), pp. 527–548. http://dx.doi.org/10.1007/s004400050248
  • [12] T. Duncan, Y. Hu and B. Pasik-Dunkan: “Stochastic calculus for fractional Brownian motion I. Theory”, SIAM J. Control Optim., Vol. 38(2), (2000), pp. 582–612. http://dx.doi.org/10.1137/S036301299834171X
  • [13] J. M. E. Guerra and D. Nualart: “The 1/H-variation of the divergence integral with respect to the fractional Brownian motion for H > 1/2 and fractional Bessel processes”, Stoch. Proc. Appl., Vol. 115, (2005), pp. 91–115. http://dx.doi.org/10.1016/j.spa.2004.07.008
  • [14] Y. Hu and P.A. Meyer: Sur les integrales multiples de Stratonovich, Séminaire de Probabilités XXII, Lecture Notes in Math., Vol. 1321, Springer-Verlag, 1988, pp. 72-81.
  • [15] Y. Hu and P.A. Meyer: “”On the approximation of Stratonovich multiple integrals”, In: S. Cambanis, J.K. Ghosh, R.L. Karandikar and P.K. Sen (Eds.): Stochastic Processes: A festschrift in honor of G. Kallianpur, Springer-Verlag, 1993, pp. 141-147.
  • [16] K. Itô: “Multiple Wiener integral”, J. Math. Soc. Japan, Vol. 3, (1951), pp. 157–169. http://dx.doi.org/10.2969/jmsj/00310157
  • [17] H.P. McKean: “Wiener’s theory of nonlinear noise”, In: Stochastic Differential Equations. Proc. SIAM-AMS, Vol. 6, (1973), pp. 191–289.
  • [18] T. Mori and H. Oodaira: “The law of the iterated logarithm for self-similar processes represented by multiple Wiener integrals”, Probab. Th. Rel. Fields, Vol. 71, (1986), pp. 367–391. http://dx.doi.org/10.1007/BF01000212
  • [19] D. Nualart: The Malliavin Calculus and Related Topics, Springer-Verlag, 1995.
  • [20] D. Nualart: “Stochastic integration with respect to fractional Brownian motion and aplications”, In: J.M. Gonzales-Barrios, J. León and A. Meda (Eds.): Stochastic Models. Contemporary Mathematics, Vol. 336, (2003), pp. 3–39.
  • [21] V. Pérez-Abreu and C. Tudor: “Multiple stochastic fractional integrals: A transfer principle for multiple stochastic fractional integrals”, Bol. Soc. Mat. Mex., Vol. 8(3), (2002), pp. 187–203.
  • [22] V. Pipiras and M. Taqqu: “Are classes of deterministic integrands for fractional Brownian motion on an interval complete?”, Bernoulli, Vol. 7, (2001), pp. 873–897. http://dx.doi.org/10.2307/3318624
  • [23] S.G. Samko, A.A. Kilbas and O.I. Marichev: Fractional Integrals and Derivatives, Gordon and Breach Science, 1993.
  • [24] J.L. Solé and F. Utzet: “Stratonovich integral and trace”, Stoch. Stoch. Rep., Vol. 29, (1990), pp. 203–220.
  • [25] E. M. Stein: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1971.
  • [26] C. Tudor: “Some properties of the sub-fractional Brownian motion”, Stochastics, (2007) (to appear).
  • [27] J.H.C. Woerner: “Variational sums and power variation: a unifying approach to model selection and estimation in semimartingale models”, Statistics and Decisions, Vol. 21, (2003), pp. 47–68. http://dx.doi.org/10.1524/stnd.21.1.47.20316
  • [28] J.H.C. Woerner: “Estimation of integrated volatility in stochastic volatility models”, Appl. Stoch. Models Bus., Vol. 21, (2005), pp. 27–44. http://dx.doi.org/10.1002/asmb.548
  • [29] M. Zakai: “Stochastic integration, trace and the skeleton of Wiener functionals”, Stochastics, Vol. 32, (1990), pp. 93–108.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-007-0001-9
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