American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[2]

Q. Feng and X. Zhang, Taylor expansions and Castell estimates for solutions of stochastic differential equations driven by rough paths, preprint,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, 1997. doi: 10.1017/CBO9780511526169.  Google Scholar

[10]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[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.  Google Scholar

[12]

T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310. doi: 10.4171/RMI/240.  Google Scholar

[13]

Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0.  Google Scholar

[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.  Google Scholar

[15]

D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.  Google Scholar

[16]

C. Reutenauer, Free Lie Algebra, vol. 7 of London Mathematical Society Monographs, New Series, Oxford University Press, 1993.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

Q. Feng and X. Zhang, Taylor expansions and Castell estimates for solutions of stochastic differential equations driven by rough paths, preprint,, , ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

S. Janson, Gaussian Hilbert Spaces, Cambridge University Press, 1997. doi: 10.1017/CBO9780511526169.  Google Scholar

[10]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992. doi: 10.1007/978-3-662-12616-5.  Google Scholar

[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.  Google Scholar

[12]

T. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310. doi: 10.4171/RMI/240.  Google Scholar

[13]

Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0.  Google Scholar

[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.  Google Scholar

[15]

D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion, Collect. Math., 53 (2002), 55-81.  Google Scholar

[16]

C. Reutenauer, Free Lie Algebra, vol. 7 of London Mathematical Society Monographs, New Series, Oxford University Press, 1993.  Google Scholar

[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.  Google Scholar

[1]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[2]

Vladimir Georgiev, Koichi Taniguchi. On fractional Leibniz rule for Dirichlet Laplacian in exterior domain. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1101-1115. doi: 10.3934/dcds.2019046
[3]

María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473
[4]

Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with non-instantaneous impulses driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2521-2541. doi: 10.3934/dcdsb.2017084
[5]

Yousef Alnafisah, Hamdy M. Ahmed. Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021031
[6]

Xinfu Chen, Bei Hu, Jin Liang, Yajing Zhang. Convergence rate of free boundary of numerical scheme for American option. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1435-1444. doi: 10.3934/dcdsb.2016004
[7]

Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$-brownian motion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1459-1502. doi: 10.3934/dcdsb.2018159
[8]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133
[9]

S. Kanagawa, K. Inoue, A. Arimoto, Y. Saisho. Mean square approximation of multi dimensional reflecting fractional Brownian motion via penalty method. Conference Publications, 2005, 2005 (Special) : 463-475. doi: 10.3934/proc.2005.2005.463
[10]

Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 615-635. doi: 10.3934/dcdsb.2018199
[11]

Wen Li, Song Wang, Volker Rehbock. A 2nd-order one-point numerical integration scheme for fractional ordinary differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 273-287. doi: 10.3934/naco.2017018
[12]

Seda İğret Araz. New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2297-2309. doi: 10.3934/dcdss.2021053
[13]

Brahim Boufoussi, Soufiane Mouchtabih. Controllability of neutral stochastic functional integro-differential equations driven by fractional brownian motion with Hurst parameter lesser than $ 1/2 $. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020096
[14]

Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157
[15]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[16]

Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23
[17]

Roberto Garrappa, Eleonora Messina, Antonia Vecchio. Effect of perturbation in the numerical solution of fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2679-2694. doi: 10.3934/dcdsb.2017188
[18]

Joseph A. Connolly, Neville J. Ford. Comparison of numerical methods for fractional differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 289-307. doi: 10.3934/cpaa.2006.5.289
[19]

Zhengyan Lin, Li-Xin Zhang. Convergence to a self-normalized G-Brownian motion. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 4-. doi: 10.1186/s41546-017-0013-8
[20]

Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (96)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences