A Milstein-type scheme without Lévy area terms for SDEs driven by fractional brownian motion
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 2, pp. 518-550.

Nous étudions dans cet article l'approximation numérique d'équations différentielles dirigées par un mouvement brownien fractionnaire (mBf) de coefficient de Hurst H > 1/3. L'algorithme effectif que nous proposons repose sur un développement au second ordre, où l'aire de Lévy est remplacée par un produit d'incréments du mBf. Nous obtenons la convergence de notre schéma en combinant des méthodes issues de la théorie des trajectoires rugueuses et des résultats sur l'approximation de l'aire de Lévy.

In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second-order Taylor expansion, where the usual Lévy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretization of the Lévy area terms.

DOI : 10.1214/10-AIHP392
Classification : 60H35, 60H07, 60H10, 65C30
Mots clés : fractional brownian motion, Lévy area, approximation schemes
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     title = {A {Milstein-type} scheme without {L\'evy} area terms for {SDEs} driven by fractional brownian motion},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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Deya, A.; Neuenkirch, A.; Tindel, S. A Milstein-type scheme without Lévy area terms for SDEs driven by fractional brownian motion. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 2, pp. 518-550. doi : 10.1214/10-AIHP392. http://archive.numdam.org/articles/10.1214/10-AIHP392/

[1] E. Alòs and D. Nualart. Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75 (2003) 129-152. | MR | Zbl

[2] F. Baudoin and L. Coutin. Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Process. Appl. 117 (2007) 550-574. | MR | Zbl

[3] C. Bender, T. Sottinen and E. Valkeila. Pricing by hedging and no-arbitrage beyond semimartingales. Finance Stoch. 12 (2008) 441-468. | MR | Zbl

[4] S. M. Berman. A law of large numbers for the maximum in a stationary Gaussian sequence. Ann. Math. Statist. 33 (1962) 93-97. | MR | Zbl

[5] T. Björk and H. Hult. A note on Wick products and the fractional Black-Scholes model. Finance Stoch. 9 (2005) 197-209. | MR | Zbl

[6] T. Cass, P. Friz and N. Victoir. Non-degeneracy of Wiener functionals arising from rough differential equations. Trans. Amer. Math. Soc. 361 (2009) 3359-3371. | MR | Zbl

[7] S. Chang, S. Li, M. Chiang, S. Hu and M. Hsyu. Fractal dimension estimation via spectral distribution function and its application to physiological signals. IEEE Trans. Biol. Eng. 54 (2007) 1895-1898.

[8] J. M. Corcuera. Power variation analysis of some integral long-memory processes. In Stochastic Analysis and Applications 219-234. F. E. Benth et al. (Eds). Abel Symposia 2. Springer, Berlin, 2007. | MR | Zbl

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

[10] N. J. Cutland, P. E. Kopp and W. Willinger. Stock price returns and the Joseph effect: A fractional version of the Black-Scholes model. In Seminar on Stochastic Analysis, Random Fields and Applications 327-351. E. Bolthausen et al. (Eds). Prog. Probab. 36. Birkhäuser, Basel, 1995. | MR | Zbl

[11] A. Davie. Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express 2 (2007) 1-40. | MR | Zbl

[12] L. Decreusefond and D. Nualart. Flow properties of differential equations driven by fractional Brownian motion. In Stochastic Differential Equations: Theory and Applications 249-262. P. H. Baxendale et al. (Eds). Interdiscip. Math. Sci. 2. World Sci. Publ., Hackensack, NJ, 2007. | MR | Zbl

[13] A. Deya and S. Tindel. Rough Volterra equations 2: Convolutional generalized integrals. Preprint, 2008. | MR | Zbl

[14] G. Denk, D. Meintrup and S. Schäffler. Transient noise simulation: Modeling and simulation of 1/f-noise. In Modeling, Simulation, and Optimization of Integrated Circuits 251-267. K. Antreich et al. (Eds). Int. Ser. Numer. Math. 146. Birkhäuser, Basel, 2001. | MR | Zbl

[15] G. Denk and R. Winkler. Modelling and simulation of transient noise in circuit simulation. Math. Comput. Model. Dyn. Syst. 13 (2007) 383-394. | MR | Zbl

[16] D. Feyel and A. De La Pradelle. Curvilinear integrals along enriched paths. Electron. J. Probab. 11 (2006) 860-892. | MR | Zbl

[17] P. Friz and N. Victoir. Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge Univ. Press, Cambridge, 2010. | MR | Zbl

[18] M. J. Garrido Atienza, P. E. Kloeden and A. Neuenkirch. Discretization of the attractor of a system driven by fractional Brownian motion. Appl. Math. Optim. 60 (2009) 151-172. | MR | Zbl

[19] P. Guasoni. No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math. Finance 16 (2006) 569-582. | MR | Zbl

[20] M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004) 86-140. | MR | Zbl

[21] M. Gubinelli. Ramification of rough paths. J. Differential Equations 248 (2010) 693-721. | MR

[22] M. Gubinelli and S. Tindel. Rough evolution equations. Ann. Probab. 38 (2010) 1-75. | MR | Zbl

[23] M. Hairer and A. Ohashi. Ergodic theory for SDEs with extrinsic memory. Ann. Probab. 35 (2007) 1950-1977. | MR | Zbl

[24] Y. Hu and D. Nualart. Rough path analysis via fractional calculus. Trans. Amer. Math. Soc. 361 (2009) 2689-2718. | MR | Zbl

[25] J. Hüsler, V. Piterbarg and O. Seleznjev. On convergence of the uniform norms for Gaussian processes and linear approximation problems. Ann. Appl. Probab. 13 (2003) 1615-1653. | MR | Zbl

[26] A. Jentzen, P. E. Kloeden and A. Neuenkirch. Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients. Numer. Math. 112 (2009) 41-64. | MR | Zbl

[27] P. E. Kloeden, A. Neuenkirch and R. Pavani. Multilevel Monte Carlo for stochastic differential equations with additive fractional noise. Ann. Oper. Res. 189 (2011) 255-276. | MR | Zbl

[28] P. E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations, 3rd edition. Springer, Berlin, 2009. | MR | Zbl

[29] S. Kou. Stochastic modeling in nanoscale physics: Subdiffusion within proteins. Ann. Appl. Statist. 2 (2008) 501-535. | MR

[30] T. Lyons and Z. Qian. System Control and Rough Paths. Clarendon Press, Oxford, 2002. | MR | Zbl

[31] Y. Mishura and G. Shevchenko. The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion. Stochastics 80 (2008) 489-511. | MR | Zbl

[32] T. Müller-Gronbach and K. Ritter. Minimal errors for strong and weak approximation of stochastic differential equations. In Monte Carlo and Quasi-Monte Carlo Methods 2006 53-82. A. Keller et al. (Eds). Springer, Berlin, 2008. | MR | Zbl

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

[34] 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 | Zbl

[35] A. Neuenkirch, I. Nourdin, A. Rößler and S. Tindel. Trees and asymptotic developments for fractional diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009) 157-174. | Numdam | MR | Zbl

[36] A. Neuenkirch, I. Nourdin and S. Tindel. Delay equations driven by rough paths. Electron. J. Probab. 13 (2008) 2031-2068. | MR | Zbl

[37] A. Neuenkirch, S. Tindel and J. Unterberger. Discretizing the Lévy area. Stochastic Process. Appl. 20 (2010) 223-254. | MR | Zbl

[38] I. Nourdin. A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one. In Sém. Probab. XLI 181-197. Lecture Notes in Math. 1934. Springer, Berlin, 2008. | MR | Zbl

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

[40] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Springer, Berlin, 2006. | MR | Zbl

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

[42] D. Odde, E. Tanaka, S. Hawkins and H. Buettner. Stochastic dynamics of the nerve growth cone and its microtubules during neurite outgrowth. Biotechnol. Bioeng. 50 (1996) 452-461.

[43] S. Tindel and J. Unterberger. The rough path associated to the multidimensional analytic fBm with any Hurst parameter. Collect. Math. 62 (2011) 197-223. | MR | Zbl

[44] J. Unterberger. Stochastic calculus for fractional Brownian motion with Hurst exponent H > 1/4: A rough path method by analytic extension. Ann. Probab. 37 (2009) 565-614. | MR | Zbl

[45] W. Wang. On a functional limit result for increments of a fractional Brownian motion. Acta Math. Hung. 93 (2001) 153-170. | MR | Zbl

[46] W. Willinger, M. S. Taqqu and V. Teverovsky. Stock market prices and long-range dependence. Finance Stoch. 3 (1999) 1-13. | Zbl

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

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