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Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion. (English) Zbl 1154.60338
Summary: We study the approximation of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $$H>1/2$$. For the mean-square error at a single point we derive the optimal rate of convergence that can be achieved by arbitrary approximation methods that are based on an equidistant discretization of the driving fractional Brownian motion. We find that there are mainly two cases: either the solution can be approximated perfectly or the best possible rate of convergence is $$n ^{- H - 1/2}$$, where $$n$$ denotes the number of evaluations of the fractional Brownian motion. In addition, we present an implementable approximation scheme that obtains the optimal rate of convergence in the latter case.

##### MSC:
 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 65C30 Numerical solutions to stochastic differential and integral equations
SimEstFBM
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##### References:
 [1] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions with formulas, graphs and mathematical tables, (1964), U.S. Department of Commerce Washington · Zbl 0171.38503 [2] Benhenni, K., Approximating integrals of stochastic processes: extensions, J. appl. probab., 35, 843-855, (1998) · Zbl 0924.62097 [3] Benth, F.E., On arbitrage-free pricing of weather derivatives based on fractional Brownian motion, Appl. math. finance, 10, 303-324, (2003) · Zbl 1087.91020 [4] Brody, D.; Syroka, J.; Zervos, M., Dynamical pricing of weather derivatives, Quant. finance, 2, 189-198, (2002) [5] Cambanis, S.; Hu, Y., Exact convergence rate of the euler – maruyama scheme, with application to sampling design, Stoch. stoch. rep., 59, 211-240, (1996) · Zbl 0868.60048 [6] Castell, F.; Gaines, J., The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations, Ann. inst. Henri Poincaré probab. stat., 32, 2, 231-250, (1996) · Zbl 0851.60054 [7] Clark, J.M.C.; Cameron, R.J., The maximum rate of convergence of discrete approximations, (), 161-171 [8] Coeurjolly, J.F., Simulation and identification of the fractional Brownian motion: a bibliographical and comparative study, J. statist. software, 5, 1-53, (2000) [9] Craigmile, P.F., Simulating a class of stationary Gaussian processes using the davies – harte algorithm, with application to long memory processes, J. time ser. anal., 24, 505-511, (2003) · Zbl 1035.68131 [10] Corcuera, J.M.; Nualart, D.; Woerner, J.H.C., Power variation of some integral long-memory processes, Bernoulli, 12, 713-735, (2006) · Zbl 1130.60058 [11] Denk, G.; Meintrup, D.; Schäffler, S., Transient noise simulation: modeling and simulation of $$1 / f$$-noise, (), 251-267 · Zbl 1043.65009 [12] Detemple, J.; Garsia, R.; Rindisbacher, M., Representation formulas for Malliavin derivatives of diffusion processes, Finance stoch., 9, 3, 349-367, (2005) · Zbl 1088.60057 [13] M. Gradinaru, I. Nourdin, Convergence of weighted power variations of fractional Brownian motion, 2007, Working paper [14] Y. Hu, D. Nualart, Differential equations driven by Hölder continuous functions of order greater than 1/2, 2006, Working Paper [15] Jolis, M., On the Wiener integral with respect to the fractional Brownian motion on an interval, J. math. anal. appl., 330, 1115-1127, (2007) · Zbl 1185.60057 [16] Kou, S.C.; Sunney Xie, X., Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule, Phys. rev. lett., 93, 18, (2004) [17] McShane, E.J., () [18] Malliavin, P., () [19] Y. Mishura, G. Shevchenko, The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion, 2007, Working paper · Zbl 1154.60046 [20] Müller-Gronbach, T., Optimal pointwise approximation of SDEs based on Brownian motion at discrete points, Ann. appl. probab., 14, 4, 1605-1642, (2004) · Zbl 1074.65010 [21] Neuenkirch, A., Optimal approximation of SDE’s with additive fractional noise, J. complexity, 22, 459-474, (2006) · Zbl 1106.65003 [22] A. Neuenkirch, Optimal approximation of stochastic differential equations with additive fractional noise, Ph.D. Thesis, TU Darmstadt. Shaker Verlag, Aachen, 2006 · Zbl 1096.65004 [23] Neuenkirch, A.; Nourdin, I., Exact rate of convergence of some approximation schemes associated to SDEs driven by a fBm, J. theor. probab., 20, 4, 871-899, (2007) · Zbl 1141.60043 [24] Newton, N.J., An asymptotically efficient difference formula for solving stochastic differential equations, Stochastics, 19, 175-206, (1986) · Zbl 0618.60053 [25] Newton, N.J., Asymptotically efficient runge – kutta methods for a class of Itô and Stratonovich equations, SIAM J. appl. math., 51, 2, 542-567, (1991) · Zbl 0724.65135 [26] I. Nourdin, A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one, Séminaire de Probabilités XLI (2007) (in press) [27] Nourdin, I.; Simon, T., On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion, Stat. probab. lett., 76, 9, 907-912, (2006) · Zbl 1091.60008 [28] Nualart, D., Stochastic calculus with respect to the fractional Brownian motion and applications, Contemp. math., 336, 3-39, (2003) · Zbl 1063.60080 [29] Nualart, D., The Malliavin calculus and related topics, (2006), Springer New York · Zbl 1099.60003 [30] D. Nualart, B. Saussereau, Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, 2005, Working paper · Zbl 1169.60013 [31] Ritter, K., Average-case analysis of numerical problems, (2000), Springer Berlin · Zbl 0949.65146 [32] Singer, P., An integrated fractional Fourier transform, J. comput. appl. math., 54, 221-237, (1994) · Zbl 0821.60086 [33] Stein, M.L., Predicting integrals of stochastic processes, Ann. appl. probab., 5, 1, 158-170, (1995) · Zbl 0820.62081 [34] Stein, M.L., Predicting integrals of random fields using observations on a lattice, Ann. statist., 23, 1975-1990, (1995) · Zbl 0856.62083 [35] Yamato, Y., Stochastic differential equations and nilpotent Lie algebras, Z. wahrscheinlichkeitstheor. verw. geb., 47, 213-229, (1979) · Zbl 0427.60069 [36] Zähle, M., Stochastic differential equations with fractal noise, Math. nachr., 278, 9, 1097-1106, (2005) · Zbl 1075.60075
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