Power variation of multiple fractional integrals

Constantin Tudor; Maria Tudor

Open Mathematics (2007)

  • Volume: 5, Issue: 2, page 358-372
  • ISSN: 2391-5455

How to cite

top

Constantin Tudor, and Maria Tudor. "Power variation of multiple fractional integrals." Open Mathematics 5.2 (2007): 358-372. <http://eudml.org/doc/269810>.

@article{ConstantinTudor2007,
abstract = {},
author = {Constantin Tudor, Maria Tudor},
journal = {Open Mathematics},
keywords = {Fractional Brownian motion; sub-fractional Brownian motion; multiple fractional Wiener-Itô integral; multiple fractional Stratonovich integral; q-variation; fractional Brownian motion},
language = {eng},
number = {2},
pages = {358-372},
title = {Power variation of multiple fractional integrals},
url = {http://eudml.org/doc/269810},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Constantin Tudor
AU - Maria Tudor
TI - Power variation of multiple fractional integrals
JO - Open Mathematics
PY - 2007
VL - 5
IS - 2
SP - 358
EP - 372
AB -
LA - eng
KW - Fractional Brownian motion; sub-fractional Brownian motion; multiple fractional Wiener-Itô integral; multiple fractional Stratonovich integral; q-variation; fractional Brownian motion
UR - http://eudml.org/doc/269810
ER -

References

top
  1. [1] E. Alòs and D. Nualart: “Stochastic integration with respect to the fractional Brownian motion”, Stoch. Stoch. Rep., Vol. 75, (2003), pp. 277–305. Zbl1028.60048
  2. [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 Zbl1075.60533
  3. [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. Zbl1059.62107
  4. [4] O.E. Barndorff-Nielsen and N. Shephard: “Realized power variation and stochastic volatility models”, Bernoulli, Vol. 9, (2003), pp. 243–265. Zbl1026.60054
  5. [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. [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 Zbl1141.91634
  7. [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 Zbl1076.60027
  8. [8] P. Caithamer: “Decoupled double stochastic fractional integrals”, Stochastics, Vol. 77, Vol. 3, (2005), pp. 205–210. Zbl1085.60031
  9. [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 Zbl1130.60058
  10. [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 Zbl0948.60022
  11. [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 Zbl0952.60043
  12. [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 Zbl0947.60061
  13. [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 Zbl1075.60056
  14. [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. [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. Zbl0798.60057
  16. [16] K. Itô: “Multiple Wiener integral”, J. Math. Soc. Japan, Vol. 3, (1951), pp. 157–169. http://dx.doi.org/10.2969/jmsj/00310157 Zbl0044.12202
  17. [17] H.P. McKean: “Wiener’s theory of nonlinear noise”, In: Stochastic Differential Equations. Proc. SIAM-AMS, Vol. 6, (1973), pp. 191–289. 
  18. [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 Zbl0562.60033
  19. [19] D. Nualart: The Malliavin Calculus and Related Topics, Springer-Verlag, 1995. Zbl0837.60050
  20. [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. Zbl1063.60080
  21. [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. Zbl1020.60050
  22. [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 Zbl1003.60055
  23. [23] S.G. Samko, A.A. Kilbas and O.I. Marichev: Fractional Integrals and Derivatives, Gordon and Breach Science, 1993. 
  24. [24] J.L. Solé and F. Utzet: “Stratonovich integral and trace”, Stoch. Stoch. Rep., Vol. 29, (1990), pp. 203–220. Zbl0706.60056
  25. [25] E. M. Stein: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1971. 
  26. [26] C. Tudor: “Some properties of the sub-fractional Brownian motion”, Stochastics, (2007) (to appear). Zbl1124.60038
  27. [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 Zbl1046.62084
  28. [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 Zbl1092.91034
  29. [29] M. Zakai: “Stochastic integration, trace and the skeleton of Wiener functionals”, Stochastics, Vol. 32, (1990), pp. 93–108. Zbl0722.60049

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.