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Convergence rates for the full Brownian rough paths with applications to limit theorems for stochastic flows. (English) Zbl 1237.60044
A quantitative version of a well-known limit theorem is presented. It says, in essence, that if one uses piecewise linear approximations to multidimensional Brownian driving signals, the resulting solutions to the (random) ODEs will converge as stochastic flows to the solution of the (Stratonovich) stochastic differential equations; that is, the solution flows and all their derivatives will convergence uniformly on compacts. Rough path theory is used to prove limit theorems relying on refined Hölder metrics on rough path spaces. The convergence for the full Brownian rough path involves five lemmas and two theorems. The paper is addressing highly specialized researchers dealing with stochastic analysis.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
Full Text: DOI
[1] A. Deya, A. Neuenkirch, S. Tindel, A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion, arXiv:1001.3344, Ann. Inst. H. Poincaré Probab. Statist., forthcoming. · Zbl 1260.60135
[2] Friz, P.; Oberhauser, H., Rough paths limits of the Wong-zakai type with a modified drift term, J. funct. anal., 256, 10, 3236-3256, (2009) · Zbl 1169.60011
[3] Friz, P.; Victoir, N., Multidimensional stochastic processes as rough paths, Cambridge stud. adv. math., vol. 120, (2010), Cambridge University Press · Zbl 1193.60053
[4] Gyöngy, I.; Shmatkov, A., Rate of convergence of Wong-zakai approximations for stochastic partial differential equations, Appl. math. optim., 54, 315-341, (2006) · Zbl 1106.60050
[5] Hu, Y.; Nualart, D., Rough path analysis via fractional calculus, Trans. amer. math. soc., 361, 5, 2689-2718, (2009) · Zbl 1175.60061
[6] Lyons, T., Differential equations driven by rough signals, Rev. mat. iberoam., 14, 2, 215-310, (1998) · Zbl 0923.34056
[7] Lyons, T.; Qian, Z., Flow of diffeomorphisms induced by a geometric multiplicative functional, Probab. theory related fields, 112, 1, 91-119, (1998) · Zbl 0918.60009
[8] Lyons, T.; Qian, Z., System control and rough paths, (2002), Oxford University Press · Zbl 1029.93001
[9] Malliavin, P., Stochastic analysis, Grundlehren math. wiss., vol. 313, (1997), Springer-Verlag Berlin · Zbl 0878.60001
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