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Taylor schemes for rough differential equations and fractional diffusions
1. | Department of Mathematics, The University of Kansas, Lawrence, Kansas, 66045, United States, United States, United States |
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. |
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. |
[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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