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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. doi: 10.1214/EJP.v172136. 
[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, (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. doi: 10.1214/08AIHP196. 
[5] 
Y. Hu, Strong and weak order of time discretization schemes of stochastic differential equations,, Séminaire de Probabilités XXX, 1626 (1996), 218. 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. 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. doi: 10.1214/15AAP1114. 
[8] 
Y. Hu and B. Øksendal, Fractional white noise calculus and applications to finance,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6 (2003), 1. 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: 10.1007/9783662126165. 
[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. doi: 10.4310/MRL.1994.v1.n4.a5. 
[12] 
T. Lyons, Differential equations driven by rough signals,, Rev. Mat. Iberoamericana, 14 (1998), 215. doi: 10.4171/RMI/240. 
[13] 
Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes,, SpringerVerlag, (2008). doi: 10.1007/9783540758730. 
[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. doi: 10.1007/s1095900700830. 
[15] 
D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion,, Collect. Math., 53 (2002), 55. 
[16] 
C. Reutenauer, Free Lie Algebra,, vol. 7 of London Mathematical Society Monographs, (1993). 
[17] 
M. Zähle, Integration with respect to fractal functions and stochastic calculus. I.,, Probab. Theory Related Fields, 111 (1998), 333. 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. doi: 10.1214/EJP.v172136. 
[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, (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. doi: 10.1214/08AIHP196. 
[5] 
Y. Hu, Strong and weak order of time discretization schemes of stochastic differential equations,, Séminaire de Probabilités XXX, 1626 (1996), 218. 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. 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. doi: 10.1214/15AAP1114. 
[8] 
Y. Hu and B. Øksendal, Fractional white noise calculus and applications to finance,, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6 (2003), 1. 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: 10.1007/9783662126165. 
[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. doi: 10.4310/MRL.1994.v1.n4.a5. 
[12] 
T. Lyons, Differential equations driven by rough signals,, Rev. Mat. Iberoamericana, 14 (1998), 215. doi: 10.4171/RMI/240. 
[13] 
Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes,, SpringerVerlag, (2008). doi: 10.1007/9783540758730. 
[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. doi: 10.1007/s1095900700830. 
[15] 
D. Nualart and A. Rascanu, Differential equations driven by fractional Brownian motion,, Collect. Math., 53 (2002), 55. 
[16] 
C. Reutenauer, Free Lie Algebra,, vol. 7 of London Mathematical Society Monographs, (1993). 
[17] 
M. Zähle, Integration with respect to fractal functions and stochastic calculus. I.,, Probab. Theory Related Fields, 111 (1998), 333. doi: 10.1007/s004400050171. 
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