×

zbMATH — the first resource for mathematics

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.

MSC:
60G22 Fractional processes, including fractional Brownian motion
60H05 Stochastic integrals
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.