Stochastic integration with respect to fractional brownian motion

Philippe Carmona; Laure Coutin; Gérard Montseny

Annales de l'I.H.P. Probabilités et statistiques (2003)

  • Volume: 39, Issue: 1, page 27-68
  • ISSN: 0246-0203

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Carmona, Philippe, Coutin, Laure, and Montseny, Gérard. "Stochastic integration with respect to fractional brownian motion." Annales de l'I.H.P. Probabilités et statistiques 39.1 (2003): 27-68. <http://eudml.org/doc/77756>.

@article{Carmona2003,
author = {Carmona, Philippe, Coutin, Laure, Montseny, Gérard},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Gaussian process; stochastic integral; Malliavin calculus; fractional integration},
language = {eng},
number = {1},
pages = {27-68},
publisher = {Elsevier},
title = {Stochastic integration with respect to fractional brownian motion},
url = {http://eudml.org/doc/77756},
volume = {39},
year = {2003},
}

TY - JOUR
AU - Carmona, Philippe
AU - Coutin, Laure
AU - Montseny, Gérard
TI - Stochastic integration with respect to fractional brownian motion
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2003
PB - Elsevier
VL - 39
IS - 1
SP - 27
EP - 68
LA - eng
KW - Gaussian process; stochastic integral; Malliavin calculus; fractional integration
UR - http://eudml.org/doc/77756
ER -

References

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  1. [1] E. Alos, O. Mazet, D. Nualart, Stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than 1/2, Stochastic Process. Appl.86 (2000) 121-139. Zbl1028.60047MR1741199
  2. [2] A. Benassi, S. Jaffard, D. Roux, Elliptic Gaussian random processes, Rev. Mat. Iberoamericana13 (1997) 19-90. Zbl0880.60053MR1462329
  3. [3] J. Beran, N. Terrin, Testing for a change of the long-memory parameter, Biometrika83 (1996) 627-638. Zbl0866.62055MR1423879
  4. [4] Z. Ciesielski, G. Kerkyacharian, B. Roynette, Quelques espaces fonctionnels associés à des processus gaussiens. (Some function spaces associated with gaussian processes), Stud. Math.107 (1993) 171-204. Zbl0809.60004MR1244574
  5. [5] F. Comte, E. Renault, Long memory continuous time models, J. Econometrics73 (1996) 101-150. Zbl0856.62104MR1410003
  6. [6] L. Coutin, Z. Qian, Stochastic analysis, rough path analysis and fractional Brownian motions, To be published in PTRF, 2000. Zbl1047.60029
  7. [7] L. Coutin, Z. Qian, Stochastic differential equations for fractional Brownian motions, C. R. Acad. Sci. Paris Sér. I Math.331 (2000) 75-80. Zbl0981.60040MR1780221
  8. [8] W. Dai, C. Heyde, Ito's formula with respect to fractional Brownian motion and its application, J. Appl. Math. Stochastic Anal.9 (1996) 439-448. Zbl0867.60029MR1429266
  9. [9] L. Decreusefond, A. Üstunel, Stochastic analysis of the fractional Brownian motion, Potential Anal.10 (1997) 177-214. Zbl0924.60034MR1677455
  10. [10] C. Dellacherie, B. Maisonneuve, P. Meyer, Probabilités et potentiel. Chapitres XVII à XXIV : Processus de Markov (fin) Compléments de calcul stochastique, Hermann, Paris, 1992. 
  11. [11] R. Dudley, R. Norvaisa, An introduction to p-variation and Young integrals, Tech. Rep. 1, Maphysto, Centre for Mathematical Physics and Stochastics, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark, 1998, Concentrated advanced course. Zbl0937.28001
  12. [12] T. Duncan, Y. Hu, B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion I. Theory, SIAM J. Control Optim.38 (2000) 582-612. Zbl0947.60061MR1741154
  13. [13] D. Feyel, A.de La Pradelle, On the approximate solution of the Stratonovitch equation, Electron. J. Probab.3 (1998). Zbl0901.60028MR1624858
  14. [14] A. Kolmogorov, Wienersche spiralen und einige andere interessante kurven im Hilbertschen raum, S. R. (Dokl.) Acad. Sci. USSR (N.S.)26 (1940) 115-118. Zbl66.0552.03MR3441JFM66.0552.03
  15. [15] N. Lebedev, Special Functions and their Applications, Dover Publications, New York, 1972, (Translated and edited by Richard A. Silverman). Zbl0271.33001MR350075
  16. [16] W. Leland, M. Taqqu, W. Willinger, D. Wilson, On the self-similar nature of Ethernet traffic, IEEE/ACM Trans. Networking2 (1994) 1-15. 
  17. [17] J. Leon, Fubini theorem for anticipating stochastic integrals in Hilbert space, Appl. Math. Optimization27 (1993) 313-327. Zbl0771.60039MR1201627
  18. [18] S. Lin, Stochastic analysis of fractional Brownian motions, Stochastics Stochastics Rep.55 (1995) 121-140. Zbl0886.60076MR1382288
  19. [19] R. Liptser, A. Shyriaev, Theory of Martingales, Mathematics and its Applications, Kluwer Academic Publishers, 1989. Zbl0728.60048MR1022664
  20. [20] T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana14 (1998) 215-310. Zbl0923.34056MR1654527
  21. [21] I. Norros, E. Valkeila, J. Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli5 (1999) 571-588. Zbl0955.60034MR1704556
  22. [22] D. Nualart, The Malliavin Calculus and Related Topics, Probability and its Applications, Springer-Verlag, New York, NY, 1995. Zbl0837.60050MR1344217
  23. [23] V. Pipiras, M.S. Taqqu, Integration questions related to fractional Brownian motion, Probab. Theory Related Fields (2000) 251-291. Zbl0970.60058MR1790083
  24. [24] N. Privault, Skorokhod stochastic integration with respect to non-adapted processes on Wiener space, Stochastics Stochastics Rep.65 (1998) 13-39. Zbl0918.60038MR1708428
  25. [25] P. Protter, Stochastic Integration and Differential Equations, Applications of Mathematics, 21, Springer-Verlag, 1992. Zbl0694.60047
  26. [26] L. Rogers, Arbitrage with fractional Brownian motion, Math. Finance7 (1997) 95-105. Zbl0884.90045MR1434408
  27. [27] F. Russo, P. Vallois, Forward, backward and symmetric stochastic integration, Probab. Theory Related Fields97 (1993) 403-421. Zbl0792.60046MR1245252
  28. [28] F. Russo, P. Vallois, The generalized covariation process and Itô formula, Stochastic Process. Appl.59 (1995) 81-104. Zbl0840.60052MR1350257
  29. [29] A.A. Ruzmaikina, Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion, J. Statist. Phys.100 (2000) 1049-1069. Zbl0970.60045MR1798553
  30. [30] S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives, Gordon & Breach Science, 1993. Zbl0818.26003MR1347689
  31. [31] D. Stroock, A Concise Introduction to the Theory of Integration, Birkhauser, 1994. Zbl0912.28001MR1267228
  32. [32] L. Young, An inequality of Hölder type, connected with Stieltjes integration, Acta Math.67 (1936) 251-282. Zbl0016.10404MR1555421
  33. [33] M. Zähle, Integration with respect to fractal functions and Stochastic Calculus, Probab. Theory Related Fields111 (1998) 333-374. Zbl0918.60037MR1640795
  34. [34] M. Zähle, On the link between fractional and stochastic calculus, in: Crauel H. (Ed.), Stochastic Dynamics, Conference on Random Dynamical Systems, Bremen, Germany, April 28–May 2, 1997, Springer, 1999, pp. 305-325, Dedicated to Ludwig Arnold on the occasion of his 60th birthday. Zbl0947.60060

Citations in EuDML Documents

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  1. Raluca M. Balan, Lp-theory for the stochastic heat equation with infinite-dimensional fractional noise
  2. Patrick Cheridito, David Nualart, Stochastic integral of divergence type with respect to fractional brownian motion with Hurst parameter H ( 0 , 1 2 )
  3. Raluca M. Balan, -theory for the stochastic heat equation with infinite-dimensional fractional noise
  4. Annie Millet, Marta Sanz-Solé, Large deviations for rough paths of the fractional brownian motion
  5. L. Decreusefond, Stochastic integration with respect to Volterra processes
  6. Mihai Gradinaru, Ivan Nourdin, Francesco Russo, Pierre Vallois, m-order integrals and generalized Itô's formula ; the case of a fractional brownian motion with any Hurst index
  7. David Nualart, Stochastic calculus with respect to fractional Brownian motion

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