Milstein’s type schemes for fractional SDEs

Milstein’s type schemes for fractional SDEs

Mihai Gradinaru and Ivan Nourdin

Source: Ann. Inst. H. Poincaré Probab. Statist. Volume 45, Number 4 (2009), 1085-1098.

Abstract

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.

Résumé

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.

First Page: Show

Primary Subjects: 60F15, 60G15, 60H05, 60H35

Keywords: Fractional Brownian motion; Weighted power variations; Stochastic differential equation; Milstein’s type scheme; Exact rate of convergence

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aihp/1257529893
Digital Object Identifier: doi:10.1214/08-AIHP196
Mathematical Reviews number (MathSciNet): MR2572165
Zentralblatt MATH identifier: 1197.60070

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References

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

Mathematical Reviews (MathSciNet): MR2248234

Digital Object Identifier: doi:10.3150/bj/1155735933

Project Euclid: euclid.bj/1155735933

[2] L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108–140.

Mathematical Reviews (MathSciNet): MR1883719

Zentralblatt MATH: 1047.60029

Digital Object Identifier: doi:10.1007/s004400100158

[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.

Mathematical Reviews (MathSciNet): MR2387018

Zentralblatt MATH: 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.

Mathematical Reviews (MathSciNet): MR2041819

[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.

Mathematical Reviews (MathSciNet): MR2144234

Digital Object Identifier: doi:10.1016/j.anihpb.2004.06.002

[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.

Mathematical Reviews (MathSciNet): MR378070

Digital Object Identifier: doi:10.1214/aop/1176996311

[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.

[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.

[10] S. J. Lin. Stochastic analysis of fractional Brownian motions. Stochastics Stochastics Rep. 55 (1995) 121–140.

Mathematical Reviews (MathSciNet): MR1382288

Zentralblatt MATH: 0886.60076

[11] T. J. Lyons. Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215–310.

Mathematical Reviews (MathSciNet): MR1654527

Zentralblatt MATH: 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.

Mathematical Reviews (MathSciNet): MR2456334

Digital Object Identifier: doi:10.1080/17442500802024892

[13] A. Neuenkirch. Optimal approximation of SDE’s with additive fractional noise. J. Complexity 22 (2006) 459–475.

Mathematical Reviews (MathSciNet): MR2246891

Zentralblatt MATH: 1106.65003

Digital Object Identifier: doi:10.1016/j.jco.2006.02.001

[14] A. Neuenkirch. Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion. Stochastic Process. Appl. 118 (2008) 2294–2333.

Mathematical Reviews (MathSciNet): MR2474352

Zentralblatt MATH: 1154.60338

Digital Object Identifier: doi:10.1016/j.spa.2008.01.002

[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.

Mathematical Reviews (MathSciNet): MR2359060

Zentralblatt MATH: 1141.60043

Digital Object Identifier: doi:10.1007/s10959-007-0083-0

[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.

[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.

Mathematical Reviews (MathSciNet): MR2483731

Zentralblatt MATH: 1148.60034

Digital Object Identifier: doi:10.1007/978-3-540-77913-1_8

[19] I. Nourdin and G. Peccati. Weighted power variations of iterated Brownian motion. Electron. J. Probab. 13 (2008) 1229–1256.

Mathematical Reviews (MathSciNet): MR2430706

[20] I. Nourdin and T. Simon. Correcting Newton–Côtes integrals by Lévy areas. Bernoulli 13 (2007) 695–711.

Mathematical Reviews (MathSciNet): MR2348747

Digital Object Identifier: doi:10.3150/07-BEJ6015

Project Euclid: euclid.bj/1186503483

[21] D. Nualart and A. Rǎsçanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55–81.

Mathematical Reviews (MathSciNet): MR1893308

[22] F. Russo and P. Vallois. Forward, backward and symmetric stochastic integration. Probab. Theory Related Fields 97 (1993) 403–421.

Mathematical Reviews (MathSciNet): MR1245252

Zentralblatt MATH: 0792.60046

Digital Object Identifier: doi:10.1007/BF01195073

[23] D. Talay. Résolution trajectorielle et analyse numérique des équations différentielles stochastiques. Stochastics 9 (1983) 275–306.

Mathematical Reviews (MathSciNet): MR707643

[24] M. Zähle. Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111 (1998) 333–374.

Mathematical Reviews (MathSciNet): MR1640795

Zentralblatt MATH: 0918.60037

Digital Object Identifier: doi:10.1007/s004400050171

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