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Strong convergence of neutral stochastic functional differential equations with two time-scales
1. | College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China |
2. | Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, UK |
The purpose of this paper is to discuss the strong convergence of neutral stochastic functional differential equations (NSFDEs) with two time-scales. The existence and uniqueness of invariant measure of the fast component is proved by using Wasserstein distance and the stability-in-distribution argument. The strong convergence between the slow component and the averaged component is also obtained by the the averaging principle in the spirit of Khasminskii's approach.
References:
[1] |
J. Bao, Q. Song, G. Yin and C. Yuan,
Ergodicity and strong limit results for two-time-scale functional stochastic differential equations, Stoc. Anal. Appl., 35 (2017), 1030-1046.
doi: 10.1080/07362994.2017.1349613. |
[2] |
J. Bao, G. Yin and C. Yuan,
Ergodicity for functional stochastic differential equations and applications, Nonlinear Anal., 98 (2014), 66-82.
doi: 10.1016/j.na.2013.12.001. |
[3] |
J. Bao, G. Yin and C.Yuan, Stationary Distributions for Retarded Stochastic Differential Equations without Dissipativity, arXiv: 1308.2018. |
[4] |
D. Blömker, M. Hairer and G. A. Pavliotis,
Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20 (2007), 1721-1744.
doi: 10.1088/0951-7715/20/7/009. |
[5] |
C.-E. Bréhier,
Strong and weak orders in averaging for SPDEs, Stochastic Process. Appl., 122 (2012), 2553-2593.
doi: 10.1016/j.spa.2012.04.007. |
[6] |
M.-F. Chen, From Markov Chains to Non-Equilibrium Particle Systems, Second Edition, World Scientific, Singapore, 2004.
doi: 10.1142/9789812562456. |
[7] |
G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, In: London Mathematical Society. Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996.
doi: 10.1017/CBO9780511662829. |
[8] |
R. D. Drive, A functional differential system of neutral type arising in a two-body problem of classical electrodynamics, in Nonlinear Differential Equations and Nonlinear Mechanics, 474-484, Academic Press, New York, 1963. |
[9] |
W. E, D. Liu and E. Vanden-Eijnden,
Analysis of multiscale methods for stochastic differential equations, Comm. Pure Appl. Math., 58 (2005), 1544-1585.
doi: 10.1002/cpa.20088. |
[10] |
A. Es-Sarhir, M. Scheutzow and O. van Gaans,
Invariant measures for stochastic functional differential equations with superlinear drift term, Differential Integral Equations, 23 (2010), 189-200.
|
[11] |
M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, second ed., New York, Springer-Verlag, 1998.
doi: 10.1007/978-1-4612-0611-8. |
[12] |
H. Fu, L. Wan and J. Liu,
Strong convergence in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, Stochastic Process. Appl., 125 (2015), 3255-3279.
doi: 10.1016/j.spa.2015.03.004. |
[13] |
D. Givon,
Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems, SIAM Multiscale Model. Simul., 6 (2007), 577-594.
doi: 10.1137/060673345. |
[14] |
D. Givon, I. G. Kevrekidis and R. Kupferman,
Strong convergence of projective integration schemes for singularly perturbed stochastic differential systems, Commun. Math. Sci., 4 (2006), 707-729.
doi: 10.4310/CMS.2006.v4.n4.a2. |
[15] |
M. Hairer, J. C. Mattingly and M. Scheutzow,
Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations, Probab. Theory Related Fields, 149 (2011), 223-259.
doi: 10.1007/s00440-009-0250-6. |
[16] |
Y. Kabanov and S. Pergamenshchikov, Two-Scale Stochastic Systems, Springer, Berlin, 2003.
doi: 10.1007/978-3-662-13242-5. |
[17] |
R. Z. Khasminskii,
On an averaging principle for Itô stochastic differential equations, Kibernetica, 4 (1968), 260-279.
|
[18] |
M. S. Kinnally and R. J. Williams,
On existence and uniqueness of stationary distributions for stochastic delay differential equations with positivity constraints, Electron. J. Probab., 15 (2010), 409-451.
doi: 10.1214/EJP.v15-756. |
[19] |
S. B. Kuksin and A. L. Piatnitski,
Khasminski-Whitman averaging for randonly perturbed KdV equations, J Math Pures Appl., 89 (2008), 400-428.
doi: 10.1016/j.matpur.2007.12.003. |
[20] |
H. J. Kushner,
Large deviations for two-time-scale diffusions, with delays, Appl. Math. Optim., 62 (2010), 295-322.
doi: 10.1007/s00245-010-9104-y. |
[21] |
H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston, MA, 1990.
doi: 10.1007/978-1-4612-4482-0. |
[22] |
D. Liu,
Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Commun. Math. Sci., 8 (2010), 999-1020.
doi: 10.4310/CMS.2010.v8.n4.a11. |
[23] |
Y. Liu and G. Yin,
Asymptotic expansions of transition densities for hybrid jump-diffusions, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 1-18.
doi: 10.1007/s10255-004-0143-5. |
[24] |
X. Mao, Stochastic Differential Equations and Applications, 2nd Ed., Horwood Publishing Limited, Chichester, 2008.
doi: 10.1533/9780857099402. |
[25] |
S.-E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, 1984. |
[26] |
M. Rei$\beta$, M. Riedle and O. van Gaans,
Delay differential equations driven by Lévy processes: Stationarity and Feller properties, Stochastic Process. Appl., 116 (2006), 1409-1432.
doi: 10.1016/j.spa.2006.03.002. |
[27] |
G. Yin and Q. Zhang, Continuous-time Markov Chains and Applications: A Two-Time-Scale Approach, 2nd Ed., Springer, New York, NY, 2013.
doi: 10.1007/978-1-4614-4346-9. |
show all references
References:
[1] |
J. Bao, Q. Song, G. Yin and C. Yuan,
Ergodicity and strong limit results for two-time-scale functional stochastic differential equations, Stoc. Anal. Appl., 35 (2017), 1030-1046.
doi: 10.1080/07362994.2017.1349613. |
[2] |
J. Bao, G. Yin and C. Yuan,
Ergodicity for functional stochastic differential equations and applications, Nonlinear Anal., 98 (2014), 66-82.
doi: 10.1016/j.na.2013.12.001. |
[3] |
J. Bao, G. Yin and C.Yuan, Stationary Distributions for Retarded Stochastic Differential Equations without Dissipativity, arXiv: 1308.2018. |
[4] |
D. Blömker, M. Hairer and G. A. Pavliotis,
Multiscale analysis for stochastic partial differential equations with quadratic nonlinearities, Nonlinearity, 20 (2007), 1721-1744.
doi: 10.1088/0951-7715/20/7/009. |
[5] |
C.-E. Bréhier,
Strong and weak orders in averaging for SPDEs, Stochastic Process. Appl., 122 (2012), 2553-2593.
doi: 10.1016/j.spa.2012.04.007. |
[6] |
M.-F. Chen, From Markov Chains to Non-Equilibrium Particle Systems, Second Edition, World Scientific, Singapore, 2004.
doi: 10.1142/9789812562456. |
[7] |
G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, In: London Mathematical Society. Lecture Note Series, vol. 229, Cambridge University Press, Cambridge, 1996.
doi: 10.1017/CBO9780511662829. |
[8] |
R. D. Drive, A functional differential system of neutral type arising in a two-body problem of classical electrodynamics, in Nonlinear Differential Equations and Nonlinear Mechanics, 474-484, Academic Press, New York, 1963. |
[9] |
W. E, D. Liu and E. Vanden-Eijnden,
Analysis of multiscale methods for stochastic differential equations, Comm. Pure Appl. Math., 58 (2005), 1544-1585.
doi: 10.1002/cpa.20088. |
[10] |
A. Es-Sarhir, M. Scheutzow and O. van Gaans,
Invariant measures for stochastic functional differential equations with superlinear drift term, Differential Integral Equations, 23 (2010), 189-200.
|
[11] |
M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, second ed., New York, Springer-Verlag, 1998.
doi: 10.1007/978-1-4612-0611-8. |
[12] |
H. Fu, L. Wan and J. Liu,
Strong convergence in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, Stochastic Process. Appl., 125 (2015), 3255-3279.
doi: 10.1016/j.spa.2015.03.004. |
[13] |
D. Givon,
Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems, SIAM Multiscale Model. Simul., 6 (2007), 577-594.
doi: 10.1137/060673345. |
[14] |
D. Givon, I. G. Kevrekidis and R. Kupferman,
Strong convergence of projective integration schemes for singularly perturbed stochastic differential systems, Commun. Math. Sci., 4 (2006), 707-729.
doi: 10.4310/CMS.2006.v4.n4.a2. |
[15] |
M. Hairer, J. C. Mattingly and M. Scheutzow,
Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations, Probab. Theory Related Fields, 149 (2011), 223-259.
doi: 10.1007/s00440-009-0250-6. |
[16] |
Y. Kabanov and S. Pergamenshchikov, Two-Scale Stochastic Systems, Springer, Berlin, 2003.
doi: 10.1007/978-3-662-13242-5. |
[17] |
R. Z. Khasminskii,
On an averaging principle for Itô stochastic differential equations, Kibernetica, 4 (1968), 260-279.
|
[18] |
M. S. Kinnally and R. J. Williams,
On existence and uniqueness of stationary distributions for stochastic delay differential equations with positivity constraints, Electron. J. Probab., 15 (2010), 409-451.
doi: 10.1214/EJP.v15-756. |
[19] |
S. B. Kuksin and A. L. Piatnitski,
Khasminski-Whitman averaging for randonly perturbed KdV equations, J Math Pures Appl., 89 (2008), 400-428.
doi: 10.1016/j.matpur.2007.12.003. |
[20] |
H. J. Kushner,
Large deviations for two-time-scale diffusions, with delays, Appl. Math. Optim., 62 (2010), 295-322.
doi: 10.1007/s00245-010-9104-y. |
[21] |
H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston, MA, 1990.
doi: 10.1007/978-1-4612-4482-0. |
[22] |
D. Liu,
Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Commun. Math. Sci., 8 (2010), 999-1020.
doi: 10.4310/CMS.2010.v8.n4.a11. |
[23] |
Y. Liu and G. Yin,
Asymptotic expansions of transition densities for hybrid jump-diffusions, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 1-18.
doi: 10.1007/s10255-004-0143-5. |
[24] |
X. Mao, Stochastic Differential Equations and Applications, 2nd Ed., Horwood Publishing Limited, Chichester, 2008.
doi: 10.1533/9780857099402. |
[25] |
S.-E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, 1984. |
[26] |
M. Rei$\beta$, M. Riedle and O. van Gaans,
Delay differential equations driven by Lévy processes: Stationarity and Feller properties, Stochastic Process. Appl., 116 (2006), 1409-1432.
doi: 10.1016/j.spa.2006.03.002. |
[27] |
G. Yin and Q. Zhang, Continuous-time Markov Chains and Applications: A Two-Time-Scale Approach, 2nd Ed., Springer, New York, NY, 2013.
doi: 10.1007/978-1-4614-4346-9. |
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