×

zbMATH — the first resource for mathematics

Convergence rates for the full Gaussian rough paths. (English. French summary) Zbl 1295.60045
The authors consider multi-dimensional Gaussian processes in the rough-paths framework and their results are established under the assumption that its covariance is of finite \(\rho\)-variation for some \(\rho\in[1,2)\). The main results are sharp a.s. convergence rates for approximations of Gaussian rough paths. As special cases prior results regarding Brownian motion and fractional Brownian motion are recovered. Even more specifically, for Wong-Zakai- (including piecewise linear and mollifier approximations) and Milstein-type approximations an a.s. convergence rate of \(k^{-(1/\rho-1/2-\epsilon)}\) for any \(\epsilon>0\) is obtained. The main tool for establishing the result is a multi-dimensional Young integration. Additionally, the article contains a brief introduction into the basic notions of Gaussian rough paths and shuffle algebras and interated integrals.

MSC:
60G15 Gaussian processes
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] T. Cass and P. Friz. Densities for rough differential equations under Hoermander’s condition. Ann. of Math. (2) 171 (2010) 2115-2141. · Zbl 1205.60105 · doi:10.4007/annals.2010.171.2115 · annals.princeton.edu
[2] L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108-140. · Zbl 1047.60029 · doi:10.1007/s004400100158
[3] A. M. Davie. Differential equations driven by rough paths: an approach via discrete approximation. Appl. Math. Res. Express. AMRX (2007) Art. ID abm009, 40. · Zbl 1163.34005
[4] A. Deya, A. Neuenkirch and S. Tindel. A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 518-550. · Zbl 1260.60135 · doi:10.1214/10-AIHP392
[5] P. Friz and S. Riedel. Convergence rates for the full Brownian rough paths with applications to limit theorems for stochastic flows. Bull. Sci. Math. 135 (2011) 613-628. · Zbl 1237.60044 · doi:10.1016/j.bulsci.2011.07.006
[6] P. Friz and N. Victoir. Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 369-413. · Zbl 1202.60058 · doi:10.1214/09-AIHP202 · eudml:241550
[7] P. Friz and N. Victoir. Multidimensional Stochastic Processes as Rough Paths . Cambridge Univ. Press, Cambridge, 2010. · Zbl 1193.60053
[8] P. Friz and N. Victoir. A note on higher dimensional \(p\)-variation. Electron. J. Probab. 16 (2011) 1880-1899. · Zbl 1244.60066 · doi:10.1214/EJP.v16-951
[9] I. Gyöngy and A. Shmatkov. Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations. Appl. Math. Optim. 54 (2006) 315-341. · Zbl 1106.60050 · doi:10.1007/s00245-006-1001-z
[10] M. Hairer. Rough stochastic PDEs. Comm. Pure Appl. Math. 64 (2011) 1547-1585. · Zbl 1229.60079 · doi:10.1002/cpa.20383
[11] K. Hara and M. Hino. Fractional order Taylor’s series and the neo-classical inequality. Bull. Lond. Math. Soc. 42 (2010) 467-477. · Zbl 1194.26027 · doi:10.1112/blms/bdq013
[12] Y. Hu and D. Nualart. Rough path analysis via fractional calculus. Trans. Amer. Math. Soc. 361 (2009) 2689-2718. · Zbl 1175.60061 · doi:10.1090/S0002-9947-08-04631-X
[13] S. Janson. Gaussian Hilbert Spaces . Cambridge Univ. Press, New York, 1997. · Zbl 0887.60009
[14] T. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 (1998) 215-310. · Zbl 0923.34056 · doi:10.4171/RMI/240 · eudml:39555
[15] T. Lyons and Z. Qian. System Control and Rough Paths . Oxford Univ. Press, New York, 2002. · Zbl 1029.93001 · doi:10.1093/acprof:oso/9780198506485.001.0001
[16] A. Neuenkirch, S. Tindel and J. Unterberger. Discretizing the fractional Lévy area. Stochastic Process. Appl. 120 (2010) 223-254. · Zbl 1185.60076 · doi:10.1016/j.spa.2009.10.007
[17] C. Reutenauer. Free Lie Algebras . Clarendon Press, New York, 1993. · Zbl 0798.17001
[18] N. Towghi. Multidimensional extension of L. C. Young’s inequality. JIPAM J. Inequal. Pure Appl. Math. 3 (2002) 13 (electronic). · Zbl 0997.26007 · jipam-old.vu.edu.au · eudml:122136
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.