Notes on the two-dimensional fractional Brownian motion. (English) Zbl 1093.60016
The authors consider a complex fractional Brownian motion with two independent fractional Brownian components having the same Hurst parameter. The aim of the paper is to use the stochastic calculus for fractional Brownian motion in order to obtain geometric properties of their two-dimensional motion as it has been made in the case of the planar Brownian motion. The key point is an analogue of the celebrated skew-product decomposition of the two-dimensional Brownian motion. From this representation some asymptotic properties of the motion are deduced. As a conclusion, it is emphasized that the study of the two-dimensional fractional Brownian motion with Hurst parameter \(H>1/2\) could seem simpler than the study of the planar Brownian motion, for which it is not possible to apply directly the ergodic theorem.
Reviewer: Yuliya S. Mishura (Kyïv)
MSC:
| 60G15 | Gaussian processes |
| 60F15 | Strong limit theorems |
| 60G18 | Self-similar stochastic processes |
| 60H05 | Stochastic integrals |
Keywords:
ergodic theorem; functionals of fractional Brownian motion; planar fractional Brownian motion; stochastic integrals; windingReferences:
| [1] | Alòs, E., Mazet, O. and Nualart, D. (2001). Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 766–801. · Zbl 1015.60047 · doi:10.1214/aop/1008956692 |
| [2] | Alòs, E. and Nualart, D. (2003). Stochastic calculus with respect to the fractional Brownian motion. Stochastics Stochastics Rep. 75 129–152. · Zbl 1028.60048 · doi:10.1080/1045112031000078917 |
| [3] | Baudoin, F. and Nualart, D. (2003). Equivalence of Volterra processes. Stochastic Process. Appl. 107 327–350. · Zbl 1075.60519 · doi:10.1016/S0304-4149(03)00088-7 |
| [4] | Decreusefond, L. and Üstünel, A. S. (1998). Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 177–214. · Zbl 0924.60034 · doi:10.1023/A:1008634027843 |
| [5] | Guerra, J. and Nualart, D. (2005). The \(1/H\)-variation of the divergence integral with respect to the fractional Brownian motion for \(H>1/2\) and fractional Bessel processes. Stochastic Process. Appl . 115 91–115. · Zbl 1075.60056 · doi:10.1016/j.spa.2004.07.008 |
| [6] | Lévy, P. (1965). Processus Stochastiques et Mouvement Brownien , 2nd ed. Gauthier–Villars, Paris. · Zbl 0248.60004 |
| [7] | Memin, M., Mishura, Y. and Valkeila, E. (2001). Inequalities for the moments of Wiener integrals with respect to fractional Brownian motion. Statist. Probab. Lett. 51 197–206. · Zbl 0983.60052 · doi:10.1016/S0167-7152(00)00157-7 |
| [8] | Messulam, P. and Yor, M. (1982). On D. Williams “pinching method” and some applications. J. London Math. Soc. 26 348–364. · Zbl 0518.60088 · doi:10.1112/jlms/s2-26.2.348 |
| [9] | Oodaira, H. (1972). On Strassen’s version of the law of the iterated logarithm for Gaussian processes. Z. Wahrsch. Verw. Gebiete 21 289–299. · Zbl 0214.17502 · doi:10.1007/BF00532259 |
| [10] | Orey, J. (1972). Growth rate of certain Gaussian processes. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 443–451. Univ. California Press, Berkeley. · Zbl 0284.60035 |
| [11] | Pipiras, V. and Taqqu, M. S. (2000). Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 121–291. · Zbl 0970.60058 · doi:10.1007/s004400000080 |
| [12] | Pitman, J. and Yor, M. (1986). Asymptotic laws of planar Brownian motion. Ann. Probab. 14 733–779. JSTOR: · Zbl 0607.60070 · doi:10.1214/aop/1176992436 |
| [13] | Pitman, J. and Yor, M. (1989). Further asymptotic laws of planar Brownian motion. Ann. Probab. 17 965–1011. JSTOR: · Zbl 0686.60085 |
| [14] | Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Springer, Berlin. · Zbl 0917.60006 |
| [15] | Rogers, L. C. G. (1997). Arbitrage with fractional Brownian motion. Math. Finance 7 95–105. · Zbl 0884.90045 · doi:10.1111/1467-9965.00025 |
| [16] | 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 |
| [17] | Shi, Z. (1998). Windings of Brownian motion and random walks in the plane. Ann. Probab. 26 112–131. · Zbl 0938.60073 · doi:10.1214/aop/1022855413 |
| [18] | Spitzer, F. (1958). Some theorems concerning 2-dimensional Brownian motion. Trans. Amer. Math. Soc. 87 187–197. · Zbl 0089.13601 · doi:10.2307/1993096 |
| [19] | Talagrand, M. (1995). Hausdorff measure of trajectories of multiparameter fractional Brownian motion. Ann. Probab. 23 767–775. JSTOR: · Zbl 0830.60034 · doi:10.1214/aop/1176988288 |
| [20] | Xiao, Y. (1996). Packing measure of the sample paths of fractional Brownian motion. Trans. Amer. Math. Soc. 348 3193–3213. · Zbl 0858.60039 · doi:10.1090/S0002-9947-96-01712-6 |
| [21] | Young, L. C. (1936). An inequality of the Hölder type connected with Stieltjes integration. Acta Math. 67 251–282. · Zbl 0016.10404 · doi:10.1007/BF02401743 |
| [22] | Zähle, M. (1998). Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111 333–374. · Zbl 0918.60037 · doi:10.1007/s004400050171 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
