Taylor schemes for rough differential equations and fractional diffusions

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November 2016, 21(9): 3115-3162. doi: 10.3934/dcdsb.2016090

Taylor schemes for rough differential equations and fractional diffusions

Yaozhong Hu ^1, , Yanghui Liu ^1, and David Nualart ^1,

1.	Department of Mathematics, The University of Kansas, Lawrence, Kansas, 66045, United States, United States, United States

Received October 2015 Revised March 2016 Published October 2016

Abstract
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In this paper, we study two variations of the time discrete Taylor schemes for rough differential equations and for stochastic differential equations driven by fractional Brownian motions. One is the incomplete Taylor scheme which excludes some terms of an Taylor scheme in its recursive computation so as to reduce the computation time. The other one is to add some deterministic terms to an incomplete Taylor scheme to improve the mean rate of convergence. Almost sure rate of convergence and $L_p$-rate of convergence are obtained for the incomplete Taylor schemes. Almost sure rate is expressed in terms of the Hölder exponents of the driving signals and the $L_p$-rate is expressed by the Hurst parameters. Both the almost sure and the $L_{p}$-convergence rates can be computed explicitly in terms of the parameters and the number of terms included in the incomplete scheme. In this way we can design the best incomplete schemes for the almost sure or the $L_p$-convergence. As in the smooth case, general Taylor schemes are always complicated to deal with. The incomplete Taylor scheme is even more sophisticated to analyze. A new feature of our approach is the explicit expression of the error functions which will be easier to study. Estimates for multiple integrals and formulas for the iterated vector fields are obtained to analyze the error functions and then to obtain the rates of convergence.

Keywords: Rough differential equations, multiple integrals, convergence rate, numerical approximation scheme, generalized Leibniz rule, fractional Brownian motion..

Mathematics Subject Classification: Primary: 60H10, 60H05, 60H07; Secondary: 65L2.

Citation: Yaozhong Hu, Yanghui Liu, David Nualart. Taylor schemes for rough differential equations and fractional diffusions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 3115-3162. doi: 10.3934/dcdsb.2016090

References:

[1]	F. Baudoin and X. Zhang, Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions, Electron. J. Probab., 17 (2012), 1-21. doi: 10.1214/EJP.v17-2136.
[2]	Q. Feng and X. Zhang, Taylor expansions and Castell estimates for solutions of stochastic differential equations driven by rough paths, preprint,, , ().
[3]	P. K. Friz and N. Victoir, Multidimentional Stochastic Processes as Rough Paths, Theory and Applications, vol. 120 of Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511845079.
[4]	M. Gradinaru and I. Nourdin, Milstein's type schemes for fractional SDEs, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 1085-1098. doi: 10.1214/08-AIHP196.
[5]	Y. Hu, Strong and weak order of time discretization schemes of stochastic differential equations, Séminaire de Probabilités XXX, Lecture Notes in Math. 1626 (1996), 218-227, Springer, Berlin. doi: 10.1007/BFb0094650.
[6]	Y. Hu, Multiple integrals and expansion of solutions of differential equations driven by rough paths and by fractional Brownian motions, Stochastics, 85 (2013), 859-916. doi: 10.1080/17442508.2012.673615.
[7]	Y. Hu, Y. Liu and D. Nualart, Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions, Ann. Appl. Probab., 26 (2016), 1147-1207. doi: 10.1214/15-AAP1114.
[8]	Y. Hu and B. Øksendal, Fractional white noise calculus and applications to finance, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6 (2003), 1-32. doi: 10.1142/S0219025703001110.
[9]	S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, 1997. doi: 10.1017/CBO9780511526169.
[10]	P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992. doi: 10.1007/978-3-662-12616-5.
[11]	T. Lyons, Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young, Math. Res. Lett., 1 (1994), 451-464. doi: 10.4310/MRL.1994.v1.n4.a5.
[12]	T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310. doi: 10.4171/RMI/240.
[13]	Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0.
[14]	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. doi: 10.1007/s10959-007-0083-0.
[15]	D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.
[16]	C. Reutenauer, Free Lie Algebra, vol. 7 of London Mathematical Society Monographs, New Series, Oxford University Press, 1993.
[17]	M. Zähle, Integration with respect to fractal functions and stochastic calculus. I., Probab. Theory Related Fields, 111 (1998), 333-374. doi: 10.1007/s004400050171.

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References:

[1]	F. Baudoin and X. Zhang, Taylor expansion for the solution of a stochastic differential equation driven by fractional Brownian motions, Electron. J. Probab., 17 (2012), 1-21. doi: 10.1214/EJP.v17-2136.
[2]	Q. Feng and X. Zhang, Taylor expansions and Castell estimates for solutions of stochastic differential equations driven by rough paths, preprint,, , ().
[3]	P. K. Friz and N. Victoir, Multidimentional Stochastic Processes as Rough Paths, Theory and Applications, vol. 120 of Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511845079.
[4]	M. Gradinaru and I. Nourdin, Milstein's type schemes for fractional SDEs, Ann. Inst. Henri Poincaré Probab. Stat., 45 (2009), 1085-1098. doi: 10.1214/08-AIHP196.
[5]	Y. Hu, Strong and weak order of time discretization schemes of stochastic differential equations, Séminaire de Probabilités XXX, Lecture Notes in Math. 1626 (1996), 218-227, Springer, Berlin. doi: 10.1007/BFb0094650.
[6]	Y. Hu, Multiple integrals and expansion of solutions of differential equations driven by rough paths and by fractional Brownian motions, Stochastics, 85 (2013), 859-916. doi: 10.1080/17442508.2012.673615.
[7]	Y. Hu, Y. Liu and D. Nualart, Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions, Ann. Appl. Probab., 26 (2016), 1147-1207. doi: 10.1214/15-AAP1114.
[8]	Y. Hu and B. Øksendal, Fractional white noise calculus and applications to finance, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6 (2003), 1-32. doi: 10.1142/S0219025703001110.
[9]	S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, 1997. doi: 10.1017/CBO9780511526169.
[10]	P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992. doi: 10.1007/978-3-662-12616-5.
[11]	T. Lyons, Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young, Math. Res. Lett., 1 (1994), 451-464. doi: 10.4310/MRL.1994.v1.n4.a5.
[12]	T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310. doi: 10.4171/RMI/240.
[13]	Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0.
[14]	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. doi: 10.1007/s10959-007-0083-0.
[15]	D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.
[16]	C. Reutenauer, Free Lie Algebra, vol. 7 of London Mathematical Society Monographs, New Series, Oxford University Press, 1993.
[17]	M. Zähle, Integration with respect to fractal functions and stochastic calculus. I., Probab. Theory Related Fields, 111 (1998), 333-374. doi: 10.1007/s004400050171.

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