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A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion. (English. French summary) Zbl 1260.60135

This paper is devoted to the numerical approximation of the following differential equation \[ Y_t=a+\sum_{i=1}^m\int_0^t\sigma^{(i)}(Y_u)dB_u^{(i)} \quad t\in [0,T],\;a\in \mathbb{R}^d, \] where \(\sigma=(\sigma^{(1)},\dots,\sigma^{(m)})\) is smooth and \(B=(B^{(1)},\dots,B^{(m)})\) is a \(m\)-dimensional fractional Brownian motion with Hurst parameter \(H>1/3\). As the Euler scheme is not appropriate for \(1/3<H<1/2\), the authors propose a scheme \(Z^n\) of Milstein type where the iterated integrals are replaced by simple products of increments. If \(1/3<\gamma<H\), they obtain an a.s. error of order \(\sqrt{\log n}\;n^{-(H-\gamma)}\) in the \(\gamma\)-Hölder norm. A first step in the proof consists in approximating \(Y\) by its Wong-Zakai approximation \(\overline{Z}^n\). In the second step, \(Z^n\) is considered as a second-order Taylor scheme for \(\overline{Z}^n\). Both steps use a theorem on Lipschitz continuity of solutions of rough differential equations and their Lévy areas.

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] E. Alòs and D. Nualart. Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75 (2003) 129-152. · Zbl 1028.60048 · doi:10.1080/1045112031000078917
[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. · Zbl 1119.60043 · doi:10.1016/j.spa.2006.09.004
[3] C. Bender, T. Sottinen and E. Valkeila. Pricing by hedging and no-arbitrage beyond semimartingales. Finance Stoch. 12 (2008) 441-468. · Zbl 1199.91170 · doi:10.1007/s00780-008-0074-8
[4] S. M. Berman. A law of large numbers for the maximum in a stationary Gaussian sequence. Ann. Math. Statist. 33 (1962) 93-97. · Zbl 0109.11803 · doi:10.1214/aoms/1177704714
[5] T. Björk and H. Hult. A note on Wick products and the fractional Black-Scholes model. Finance Stoch. 9 (2005) 197-209. · Zbl 1092.91021 · doi:10.1007/s00780-004-0144-5
[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. · Zbl 1175.60034 · doi:10.1090/S0002-9947-09-04677-7
[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. · Zbl 1136.60035 · doi:10.1007/978-3-540-70847-6_9
[9] L. Coutin and Z. Qian. Stochastic rough path analysis and fractional Brownian motion. Probab. Theory Related. Fields 122 (2002) 108-140. · Zbl 1047.60029 · doi:10.1007/s004400100158
[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. · Zbl 0827.60021
[11] A. Davie. Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express 2 (2007) 1-40. · Zbl 1163.34005 · doi:10.1093/amrx/abm009
[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. · Zbl 1135.60034
[13] A. Deya and S. Tindel. Rough Volterra equations 2: Convolutional generalized integrals. Preprint, 2008. · Zbl 1223.60031 · doi:10.1016/j.spa.2011.05.003
[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. · Zbl 1043.65009 · doi:10.1007/978-3-0348-8065-7_16
[15] G. Denk and R. Winkler. Modelling and simulation of transient noise in circuit simulation. Math. Comput. Model. Dyn. Syst. 13 (2007) 383-394. · Zbl 1117.93305 · doi:10.1080/13873950500064400
[16] D. Feyel and A. de La Pradelle. Curvilinear integrals along enriched paths. Electron. J. Probab. 11 (2006) 860-892. · Zbl 1110.60031 · doi:10.1214/EJP.v11-356
[17] P. Friz and N. Victoir. Multidimensional Stochastic Processes as Rough Paths: Theory and Applications . Cambridge Univ. Press, Cambridge, 2010. · Zbl 1193.60053 · doi:10.1017/CBO9780511845079
[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. · Zbl 1180.93095 · doi:10.1007/s00245-008-9062-9
[19] P. Guasoni. No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math. Finance 16 (2006) 569-582. · Zbl 1133.91421 · doi:10.1111/j.1467-9965.2006.00283.x
[20] M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004) 86-140. · Zbl 1058.60037 · doi:10.1016/j.jfa.2004.01.002
[21] M. Gubinelli. Ramification of rough paths. J. Differential Equations 248 (2010) 693-721. · Zbl 1315.60065 · doi:10.1016/j.jde.2009.11.015
[22] M. Gubinelli and S. Tindel. Rough evolution equations. Ann. Probab. 38 (2010) 1-75. · Zbl 1193.60070 · doi:10.1214/08-AOP437
[23] M. Hairer and A. Ohashi. Ergodic theory for SDEs with extrinsic memory. Ann. Probab. 35 (2007) 1950-1977. · Zbl 1129.60052 · doi:10.1214/009117906000001141
[24] Y. Hu and D. Nualart. Rough path analysis via fractional calculus. Trans. Amer. Math. Soc. 361 (2009) 2689-2718. · Zbl 1175.60061 · doi:10.1090/S0002-9947-08-04631-X
[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. · Zbl 1038.60040 · doi:10.1214/aoap/1069786514
[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. · Zbl 1163.65003 · doi:10.1007/s00211-008-0200-8
[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. · Zbl 1235.60064 · doi:10.1007/s10479-009-0663-8
[28] P. E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations , 3rd edition. Springer, Berlin, 2009. · Zbl 0752.60043
[29] S. Kou. Stochastic modeling in nanoscale physics: Subdiffusion within proteins. Ann. Appl. Statist. 2 (2008) 501-535. · Zbl 1400.62272 · doi:10.1214/07-AOAS149
[30] T. Lyons and Z. Qian. System Control and Rough Paths . Clarendon Press, Oxford, 2002. · Zbl 1029.93001 · doi:10.1093/acprof:oso/9780198506485.001.0001
[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. · Zbl 1154.60046 · doi:10.1080/17442500802024892
[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. · Zbl 1140.65305 · doi:10.1007/978-3-540-74496-2_4
[33] A. Neuenkirch. Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion. Stochastic Process. Appl. 118 (2008) 2294-2333. · Zbl 1154.60338 · doi:10.1016/j.spa.2008.01.002
[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. · Zbl 1141.60043 · doi:10.1007/s10959-007-0083-0
[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. · Zbl 1172.60017 · doi:10.1214/07-AIHP159
[36] A. Neuenkirch, I. Nourdin and S. Tindel. Delay equations driven by rough paths. Electron. J. Probab. 13 (2008) 2031-2068. · Zbl 1190.60046 · doi:10.1214/EJP.v13-575
[37] A. Neuenkirch, S. Tindel and J. Unterberger. Discretizing the Lévy area. Stochastic Process. Appl. 20 (2010) 223-254. · Zbl 1185.60076 · doi:10.1016/j.spa.2009.10.007
[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. · Zbl 1148.60034 · doi:10.1007/978-3-540-77913-1_8
[39] I. Nourdin and T. Simon. Correcting Newton-Cotes integrals by Lévy areas. Bernoulli 13 (2007) 695-711. · Zbl 1132.60047 · doi:10.3150/07-BEJ6015
[40] D. Nualart. The Malliavin Calculus and Related Topics , 2nd edition. Springer, Berlin, 2006. · Zbl 1099.60003
[41] D. Nualart and A. Rǎşcanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002) 55-81. · Zbl 1018.60057
[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. · Zbl 1220.60022 · doi:10.1007/s13348-010-0021-9
[44] J. Unterberger. Stochastic calculus for fractional Brownian motion with Hurst exponent H &gt; 1/4: A rough path method by analytic extension. Ann. Probab. 37 (2009) 565-614. · Zbl 1172.60007 · doi:10.1214/08-AOP413
[45] W. Wang. On a functional limit result for increments of a fractional Brownian motion. Acta Math. Hung. 93 (2001) 153-170. · Zbl 1001.60033 · doi:10.1023/A:1013829802476
[46] W. Willinger, M. S. Taqqu and V. Teverovsky. Stock market prices and long-range dependence. Finance Stoch. 3 (1999) 1-13. · Zbl 0924.90029 · doi:10.1007/s007800050049
[47] M. Zähle. Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111 (1998) 333-374. · Zbl 0918.60037 · doi:10.1007/s004400050171
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