Milstein's type schemes for fractional SDEs
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 1085-1098.

On étudie la vitesse exacte de convergence de certains schémas d'approximation associés à des équations différentielles stochastiques scalaires dirigées par le mouvement brownien fractionnaire B. On utilise le comportement asymptotique des variations à poids de B, et la limite de l'erreur entre la solution et son approximation est calculée de façon explicite.

Weighted power variations of fractional brownian motion B are used to compute the exact rate of convergence of some approximating schemes associated to one-dimensional stochastic differential equations (SDEs) driven by B. The limit of the error between the exact solution and the considered scheme is computed explicitly.

DOI : https://doi.org/10.1214/08-AIHP196
Classification : 60F15,  60G15,  60H05,  60H35
Mots clés : fractional brownian motion, weighted power variations, stochastic differential equation, Milstein's type scheme, exact rate of convergence
@article{AIHPB_2009__45_4_1085_0,
     author = {Gradinaru, Mihai and Nourdin, Ivan},
     title = {Milstein's type schemes for fractional SDEs},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1085--1098},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {4},
     year = {2009},
     doi = {10.1214/08-AIHP196},
     zbl = {1197.60070},
     mrnumber = {2572165},
     language = {en},
     url = {archive.numdam.org/item/AIHPB_2009__45_4_1085_0/}
}
Gradinaru, Mihai; Nourdin, Ivan. Milstein's type schemes for fractional SDEs. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 1085-1098. doi : 10.1214/08-AIHP196. http://archive.numdam.org/item/AIHPB_2009__45_4_1085_0/

[1] J. M. Corcuera, D. Nualart and J. H. C. Woerner. Power variation of some integral fractional processes. Bernoulli 12 (2006) 713-735. | MR 2248234 | Zbl 1130.60058

[2] L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108-140. | MR 1883719 | Zbl 1047.60029

[3] A. M. Davie. Differential equations driven by rough paths: An approach via discrete approximation. AMRX Appl. Math. Res. Express 2007 (2007) abm009, 1-40. | MR 2387018 | Zbl 1163.34005

[4] M. Gradinaru and I. Nourdin. Approximation at first and second order of the m-variation of the fractional Brownian motion. Electron. J. Probab. 8 (2003) 1-26. | MR 2041819 | Zbl 1063.60079

[5] M. Gradinaru, I. Nourdin, F. Russo and P. Vallois. m-order integrals and generalized Itô's formula; the case of a fractional Brownian motion with any Hurst index. Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 781-806. | Numdam | MR 2144234 | Zbl 1083.60045

[6] J. Jacod. Limit of random measures associated with the increments of a Brownian semimartingale. LPMA, preprint (revised version), 1994.

[7] R. Klein and E. Giné. On quadratic variation of processes with Gaussian increments. Ann. Probab. 3 (1975) 716-721. | MR 378070 | Zbl 0318.60031

[8] T. G. Kurtz and P. Protter. Wong-Zakai corrections, random evolutions and simulation schemes for SDEs. In Stochastic Analysis 331-346. Academic Press, Boston, MA, 1991. | MR 1119837 | Zbl 0762.60047

[9] J. R. León and C. Ludeña. Limits for weighted p-variations and likewise functionals of fractional diffusions with drift. Stochastic Process. Appl. 117 (2007) 271-296. | MR 2290877 | Zbl 1110.60023

[10] S. J. Lin. Stochastic analysis of fractional Brownian motions. Stochastics Stochastics Rep. 55 (1995) 121-140. | MR 1382288 | Zbl 0886.60076

[11] T. J. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215-310. | MR 1654527 | Zbl 0923.34056

[12] Y. Mishura and G. Shevchenko. The rate of convergence of Euler approximations for solutions of stochastic differential equations driven by fractional Brownian motion. Stochastics. To appear. Available at arXiv:0705.1773. | MR 2456334 | Zbl 1154.60046

[13] A. Neuenkirch. Optimal approximation of SDE's with additive fractional noise. J. Complexity 22 (2006) 459-475. | MR 2246891 | Zbl 1106.65003

[14] A. Neuenkirch. Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion. Stochastic Process. Appl. 118 (2008) 2294-2333. | MR 2474352 | Zbl 1154.60338

[15] A. Neuenkirch and I. Nourdin. Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion. J. Theoret. Probab. 20 (2007) 871-899. | MR 2359060 | Zbl 1141.60043

[16] I. Nourdin, D. Nualart, C. Tudor. Central and non-central limit theorem for weighted power variation of fractional Brownian motion, 2007. Available at arXiv:0710.5639.

[17] I. Nourdin. Schémas d'approximation associés à une équation différentialle dirigée par une fonction hölderienne; cas du mouvement brownien fractionnaire. C. R. Acad. Sci. Paris, Ser. I 340 (2005) 611-614. | MR 2138713 | Zbl 1075.60073

[18] I. Nourdin. A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one. Sém. Probab. XLI (2008) 181-197. | MR 2483731 | Zbl 1148.60034

[19] I. Nourdin and G. Peccati. Weighted power variations of iterated Brownian motion. Electron. J. Probab. 13 (2008) 1229-1256. | MR 2430706 | Zbl pre05636538

[20] I. Nourdin and T. Simon. Correcting Newton-Côtes integrals by Lévy areas. Bernoulli 13 (2007) 695-711. | MR 2348747 | Zbl 1132.60047

[21] D. Nualart and A. Rǎsçanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55-81. | MR 1893308 | Zbl 1018.60057

[22] F. Russo and P. Vallois. Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 (1993) 403-421. | MR 1245252 | Zbl 0792.60046

[23] D. Talay. Résolution trajectorielle et analyse numérique des équations différentielles stochastiques. Stochastics 9 (1983) 275-306. | MR 707643 | Zbl 0512.60041

[24] M. Zähle. Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111 (1998) 333-374. | MR 1640795 | Zbl 0918.60037