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Discretizing the fractional Lévy area. (English) Zbl 1185.60076
Summary: We give sharp bounds for the Euler discretization of the Lévy area associated to a \(d\)-dimensional fractional Brownian motion. We show that there are three different regimes for the exact root mean square convergence rate of the Euler scheme, depending on the Hurst parameter \(H\in (1/4,1)\). For \(H<3/4\) the exact convergence rate is \(n^{-2H+1/2}\), where \(n\) denotes the number of the discretization subintervals, while for \(H=3/4\) it is \(n^{-1}\sqrt{\log(n)}\) and for \(H>3/4\) the exact rate is \(n-1\). Moreover, we also show that a trapezoidal scheme converges (at least) with the rate \(n-2H+1/2\). Finally, we derive the asymptotic error distribution of the Euler scheme. For \(H\leq 3/4\) one obtains a Gaussian limit, while for \(H>3/4\) the limit distribution is of Rosenblatt type.

MSC:
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H07 Stochastic calculus of variations and the Malliavin calculus
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
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[1] Baudoin, F.; Coutin, L., Operators associated with a stochastic differential equation driven by fractional Brownian motions, Stoch. proc. appl., 117, 5, 550-574, (2007) · Zbl 1119.60043
[2] Begyn, A., Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes, Bernoulli, 13, 3, 712-753, (2007) · Zbl 1143.60030
[3] Breton, J.-C.; Nourdin, I., Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion, Electron. comm. probab., 13, 482-493, (2008) · Zbl 1189.60084
[4] Caruana, M.; Friz, P., Partial differential equations driven by rough paths, J. differential equation, 247, 1, 140-173, (2009) · Zbl 1167.35386
[5] T. Cass, P. Friz, N. Victoir, Non-degeneracy of Wiener functionals arising from rough differential equations, Trans. Amer. Math. Soc. 361 (2009) 3359-3371 · Zbl 1175.60034
[6] Coutin, L.; Qian, Z., Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. theory related fields, 122, 1, 108-140, (2002) · Zbl 1047.60029
[7] Davie, A., Differential equations driven by rough paths: an approach via discrete approximation, Appl. math. res. express., 2, 40 pp, (2007) · Zbl 1163.34005
[8] P. Friz, N. Victoir, Multidimensional Stochastic Processes as Rough Paths, Cambridge University Press (in press) · Zbl 1193.60053
[9] Gubinelli, M., Controlling rough paths, J. funct. anal., 216, 1, 86-140, (2004) · Zbl 1058.60037
[10] M. Gradinaru, I. Nourdin, Milstein’s type scheme for fractional SDEs, Ann. Inst. H. Poincaré Probab. Statist. (in press) · Zbl 1197.60070
[11] M. Gubinelli, S. Tindel, Rough evolution equations, preprint (arXiv:0803.0552v1). Ann. Probab. (2008) (in press)
[12] Hairer, M., Ergodicity of stochastic differential equations driven by fractional Brownian motion, Ann. probab., 33, 2, 703-758, (2005) · Zbl 1071.60045
[13] Kahane, J.-P., Some random series of functions, (1985), Cambridge University Press
[14] Kloeden, P.E.; Platen, E., Numerical solution of stochastic differential equations, (1999), Springer · Zbl 0701.60054
[15] Le Bellac, M., ()
[16] Lyons, T.; Qian, Z., System control and rough paths, (2002), Oxford University Press · Zbl 1029.93001
[17] Milstein, G.N.; Tretyakov, M.V., Stochastic numerics for mathematical physics, (2004), Springer · Zbl 1085.60004
[18] Mishura, Y.; Shevchenko, G., The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion, Stochastics, 80, 5, 489-511, (2008) · Zbl 1154.60046
[19] Neuenkirch, A., Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion, Stoch. proc. appl., 118, 12, 2294-2333, (2008) · Zbl 1154.60338
[20] Neuenkirch, A.; Nourdin, I., Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion, J. theoret. probab., 20, 4, 871-899, (2007) · Zbl 1141.60043
[21] Neuenkirch, A.; Nourdin, I.; Tindel, S., Delay equations driven by rough paths, Electron. J. probab., 13, 2031-2068, (2008) · Zbl 1190.60046
[22] Nourdin, I., A change of variable formula for the 2D fractional Brownian motion of Hurst index bigger or equal to 1/4, J. funct. anal., 256, 2303-2320, (2009) · Zbl 1162.60010
[23] I. Nourdin, D. Nualart, C. Tudor, Central and non-central limit theorems for weighted power variations of fractional Brownian motion (2008) preprint (arXiv:0710.5639v2) · Zbl 1221.60031
[24] Nualart, D.; Peccati, G., Central limit theorems for sequences of multiple stochastic integrals, Ann. probab., 33, 1, 177-193, (2005) · Zbl 1097.60007
[25] S. Tindel, J. Unterberger, The rough path associated to the multidimensional analytic fBm with any Hurst parameter, (2008) preprint (arXiv:0810.1408v1) · Zbl 1220.60022
[26] Unterberger, J., Stochastic calculus for fractional Brownian motion with Hurst exponent \(H > 1 / 4\): A rough path method by analytic extension, Ann. probab., 37, 2, 565-614, (2009) · Zbl 1172.60007
[27] J. Unterberger, A central limit theorem for the rescaled Lévy area of two-dimensional fractional Brownian motion with Hurst index \(H < 1 / 4\) (2008) preprint (arXiv:0808.3458)
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