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The rough path associated to the multidimensional analytic fBm with any Hurst parameter. (English) Zbl 1220.60022
Summary: We consider a complex-valued \(d\)-dimensional fractional Brownian motion defined on the closure of the complex upper half-plane, called analytic fractional Brownian motion and denoted by \(\Gamma \). This process has been introduced by J. Unterberger [Ann. Probab. 37, No. 2, 565–614 (2009; Zbl 1172.60007)], and both its real and imaginary parts, restricted to the real axis, are usual fractional Brownian motions. The current note is devoted to prove that a rough path based on \(\Gamma \) can be constructed for any value of the Hurst parameter in \((0, 1/2)\). We also show how to solve differential equations driven by \(\Gamma \) in a neighborhood of 0 of the complex upper half-plane, by means of elementary arguments.

60G22 Fractional processes, including fractional Brownian motion
60H05 Stochastic integrals
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI
[1] Ahlfors L.: Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. McGraw-Hill Book Co., New York (1966) · Zbl 0154.31904
[2] Baudoin F., Coutin L.: Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stoch. Process. Appl. 117, 550–574 (2007) · Zbl 1119.60043 · doi:10.1016/j.spa.2006.09.004
[3] Chang S., Li S., Chiang M., Hu S., Hsyu M.: Fractal dimension estimation via spectral distribution function and its application to physiological signals. IEEE Trans. Biol. Eng. 54, 1895–1898 (2007) · doi:10.1109/TBME.2007.894731
[4] Coutin L., Qian Z.: Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122, 108–140 (2002) · Zbl 1047.60029 · doi:10.1007/s004400100158
[5] Darses, S., Nourdin, I., Nualart, D.: Limit theorems for nonlinear functionals of Volterra processes via white noise analysis. arXiv:0901.1401 (preprint 2009) · Zbl 1225.60041
[6] Friz P., Victoir N.: Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge University Press, Cambridge (2009) · Zbl 1193.60053
[7] Garsia, A.: Continuity properties of Gaussian processes with multidimensional time parameter. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif. 1970/1971), II, Probability theory, pp. 369–374. University California Press, Berkeley (1972)
[8] Gubinelli M.: Controlling rough paths. J. Funct. Anal. 216, 86–140 (2004) · Zbl 1058.60037 · doi:10.1016/j.jfa.2004.01.002
[9] Gubinelli M.: Ramification of rough paths. J. Differ. Equ. 248, 693–721 (2010) · Zbl 1315.60065 · doi:10.1016/j.jde.2009.11.015
[10] Gubinelli M., Tindel S.: Rough evolution equations. Ann. Probab. 38, 1–75 (2010) · Zbl 1193.60070 · doi:10.1214/08-AOP437
[11] Inahama, Y.: Laplace approximation for rough differential equation driven by fractional Brownian motion. arXiv:1004.1478 (preprint, 2009) · Zbl 1273.60043
[12] Kou, S., Sunney-Xie, X.: Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule. Phys. Rev. Lett. 93 (2004)
[13] Lyons T., Qian Z.: System Control and Rough Paths. Oxford University Press, Oxford (2002) · Zbl 1029.93001
[14] Neuenkirch A., Nourdin I., Rößler A., Tindel S.: Trees and asymptotic expansions for fractional stochastic differential equations. Ann. Inst. Henri Poincaré Probab. Stat. 45, 157–174 (2009) · Zbl 1172.60017 · doi:10.1214/07-AIHP159
[15] Odde D., Tanaka E., Hawkins S., Buettner H.: Stochastic dynamics of the nerve growth cone and its microtubules during neurite outgrowth. Biotechnol. Bioeng. 50, 452–461 (1996) · doi:10.1002/(SICI)1097-0290(19960520)50:4<452::AID-BIT13>3.0.CO;2-L
[16] Unterberger J.: Stochastic calculus for fractional Brownian motion with Hurst exponent H larger than 1/4: a rough path method by analytic extension. Ann. Probab. 37, 565–614 (2009) · Zbl 1172.60007 · doi:10.1214/08-AOP413
[17] Unterberger, J.: A central limit theorem for the rescaled Lévy area of two-dimensional Brownian motion with Hurst index H &lt; 1/4. arXiv:0808.3458 (preprint 2008)
[18] Unterberger J.: A stochastic calculus for multidimensional fractional Brownian motion with arbitrary Hurst index. Stoch. Proc. Appl. 120(8), 1444–1472 (2010) · Zbl 1221.05062 · doi:10.1016/j.spa.2010.04.001
[19] Unterberger J.: Hölder-continuous rough paths by Fourier normal ordering. Comm. Math. Phys. 298(1), 1–36 (2010) · Zbl 1221.46047 · doi:10.1007/s00220-010-1064-1
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