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Stochastic calculus for fractional Brownian motion with Hurst exponent \(H>\frac 1 4 \): A rough path method by analytic extension. (English) Zbl 1172.60007

The article is devoted to the integration with respect to d-dimensional fractional Brownian motion. For the Hurst parameter \(\alpha\in (\frac{1}{4},\frac{1}{2})\) the author proposes to use the geometric rough paths theory. To do this an analytic extension of fractional Brownian motion on upper half-plane in \(C\) is constructed. It is proved that the iterated integrals from this analytic process converge to a Lévy area for fractional Brownian motion.

MSC:

60G15 Gaussian processes
60G17 Sample path properties
60H05 Stochastic integrals
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[1] Abramowitz, M., Stegun, A., Danos, M. and Rafelski, J. (1984). Handbook of Mathematical Functions . Harri Deutsch, Frankfurt.
[2] Adler, R. J. (1981). The Geometry of Random Fields . Wiley, Chichester. · Zbl 0478.60059
[3] Baudoin, F. and Coutin, L. (2005). Étude en temps petit des solutions d’EDS conduites par des mouvements browniens fractionnaires. C. R. Math. Acad. Sci. Paris 341 39-42. · Zbl 1073.60061 · doi:10.1016/j.crma.2005.05.010
[4] Cheridito, P. and Nualart, D. (2005). Stochastic integral of divergence type with respect to fractional Brownian motion with Hurst parameter H \in (0, \frac{1}{2} ). Ann. Inst. H. Poincaré Probab. Statist. 41 1049-1081. · Zbl 1083.60027 · doi:10.1016/j.anihpb.2004.09.004
[5] Coutin, L. and Qian, Z. (2002). Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 108-140. · Zbl 1047.60029 · doi:10.1007/s004400100158
[6] Decreusefond, L. and Üstünel, A. S. (1999). Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 177-214. · Zbl 0924.60034 · doi:10.1023/A:1008634027843
[7] Dzhaparidze, K. and van Zanten, H. (2004). A series expansion of fractional Brownian motion. Probab. Theory Related Fields 130 39-55. · Zbl 1059.60048 · doi:10.1007/s00440-003-0310-2
[8] Feyel, D. and de la Pradelle, A. (2001). The FBM Itô’s formula through analytic continuation. Electron. J. Probab. 6 22. · Zbl 1008.60074
[9] Gradinaru, M., Nourdin, I., Russo, F. and Vallois, P. (2005). m -order integrals and generalized Itô’s formula: The case of a fractional Brownian motion with any Hurst index. Ann. Inst. H. Poincaré Probab. Statist. 41 781-806. · Zbl 1083.60045 · doi:10.1016/j.anihpb.2004.06.002
[10] Kahane, J.-P. (1985). Some Random Series of Functions , 2nd ed. Cambridge Studies in Advanced Mathematics 5 . Cambridge Univ. Press, Cambridge. · Zbl 0571.60002
[11] Kühn, T. and Linde, W. (2002). Optimal series representation of fractional Brownian sheets. Bernoulli 8 669-696. · Zbl 1012.60074
[12] Lejay, A. (2003). An introduction to rough paths. In Séminaire de Probabilités XXXVII. Lecture Notes in Mathematics 1832 1-59. Springer, Berlin. · Zbl 1041.60051
[13] Ledoux, M., Lyons, T. and Qian, Z. (2002). Lévy area of Wiener processes in Banach spaces. Ann. Probab. 30 546-578. · Zbl 1016.60071 · doi:10.1214/aop/1023481002
[14] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 215-310. · Zbl 0923.34056 · doi:10.4171/RMI/240
[15] Lyons, T. and Qian, Z. (2002). System Control and Rough Paths . Oxford Univ. Press, Oxford. · Zbl 1029.93001 · doi:10.1093/acprof:oso/9780198506485.001.0001
[16] Nualart, D. (1995). The Malliavin Calculus and Related Topics . Springer, New York. · Zbl 0837.60050
[17] Pipiras, V. and Taqqu, M. S. (2000). Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 251-291. · Zbl 0970.60058 · doi:10.1007/s004400000080
[18] Russo, F. and Vallois, P. (1993). Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 403-421. · Zbl 0792.60046 · doi:10.1007/BF01195073
[19] Russo, F. and Vallois, P. (2000). Stochastic calculus with respect to continuous finite quadratic variation processes. Stochastics Stochastics Rep. 70 1-40. · Zbl 0981.60053
[20] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes . Springer, New York. · Zbl 0862.60002
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