September  2022, 27(9): 4725-4747. doi: 10.3934/dcdsb.2021249

Weak convergence of delay SDEs with applications to Carathéodory approximation

1. 

Department of Mathematics, VNU University of Science, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, 084, Vietnam

2. 

Department of Foundation, Academy of Cryptography Techniques, 141 Chien Thang, Thanh Tri, Hanoi, 084, Vietnam

* Corresponding author: Nguyen Tien Dung

Received  May 2020 Revised  June 2021 Published  September 2022 Early access  October 2021

Fund Project: N. T. Dung and H. T. P. Thao are supported by the Vietnam National University, Hanoi under grant number QG.20.21. T. C. Son, N. V. Tan, T. M. Cuong and P. D. Tung are supported by Viet Nam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.03-2019.08

In this paper, we consider a fundamental class of stochastic differential equations with time delays. Our aim is to investigate the weak convergence with respect to delay parameter of the solutions. Based on the techniques of Malliavin calculus, we obtain an explicit estimate for the rate of convergence. An application to the Carathéodory approximation scheme of stochastic differential equations is provided as well.

Citation: Ta Cong Son, Nguyen Tien Dung, Nguyen Van Tan, Tran Manh Cuong, Hoang Thi Phuong Thao, Pham Dinh Tung. Weak convergence of delay SDEs with applications to Carathéodory approximation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (9) : 4725-4747. doi: 10.3934/dcdsb.2021249
References:
[1]

V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function, Probab. Theory Related Fields, 104 (1996), 43-60.  doi: 10.1007/BF01303802.

[2]

D. R. Bell and S. E. A. Mohammed, On the solution of stochastic ordinary differential equations via small delays, Stochastics Stochastics Rep., 28 (1989), 293-299.  doi: 10.1080/17442508908833598.

[3]

M. Benabdallah and M. Bourza, Carathéodory approximate solutions for a class of perturbed stochastic differential equations with reflecting boundary, Stoch. Anal. Appl., 37 (2019), 936-954.  doi: 10.1080/07362994.2019.1623049.

[4]

E. BuckwarR. KuskeS.-E. Mohammed and T. Shardlow, Weak convergence of the Euler scheme for stochastic differential delay equations, LMS J. Comput. Math., 11 (2008), 60-99.  doi: 10.1112/S146115700000053X.

[5]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, , McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.

[6]

N. T. Dung and T. C. Son, Lipschitz continuity in the Hurst index of the solutions of fractional stochastic Volterra integro-differential equations, Submitted.

[7]

F. Faizullah, A note on the Carathéodory approximation scheme for stochastic differential equations under G-Brownian motion, Z. Naturforsch., 67a (2012), 699-704.  doi: 10.5560/zna.2012-0079.

[8]

M. Ferrante and C. Rovira, Convergence of delay differential equations driven by fractional Brownian motion, J. Evol. Equ., 10 (2010), 761-783.  doi: 10.1007/s00028-010-0069-8.

[9]

B. Jourdain and A. Kohatsu-Higa, A review of recent results on approximation of solutions of stochastic differential equations, Stochastic Analysis with Financial Applications, 121–144, Progr. Probab., 65, Birkhäuser/Springer Basel AG, Basel, (2011). doi: 10.1007/978-3-0348-0097-6_9.

[10]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.

[11]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[12]

K. Liu, Carathéodory approximate solutions for a class of semilinear stochastic evolution equations with time delays, J. Math. Anal. Appl., 220 (1998), 349-364.  doi: 10.1006/jmaa.1997.5889.

[13]

W. Mao, L. Hu and X. Mao, Approximate solutions for a class of doubly perturbed stochastic differential equations, Adv. Difference Equ., (2018), Paper No. 37, 17 pp. doi: 10.1186/s13662-018-1490-5.

[14]

X. Mao, Approximate solutions for a class of stochastic evolution equations with variable delays, Numer. Funct. Anal. Optim., 12 (1991), 525–533 (1992). doi: 10.1080/01630569108816448.

[15]

X. Mao, Approximate solutions for a class of stochastic evolution equations with variable delays. II, Numer. Funct. Anal. Optim., 15 (1994), 65-76.  doi: 10.1080/01630569408816550.

[16]

X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.

[17]

D. Nualart, The Malliavin Calculus and Related Topics, Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006.

[18]

B. G. Pachpatte, Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering, 197. Academic Press, Inc., San Diego, CA, 1998.

[19]

J. Turo, Carathéodory approximation solutions to a class of stochastic functional-differential equations, Appl. Anal., 61 (1996), 121-128.  doi: 10.1080/00036819608840450.

show all references

References:
[1]

V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function, Probab. Theory Related Fields, 104 (1996), 43-60.  doi: 10.1007/BF01303802.

[2]

D. R. Bell and S. E. A. Mohammed, On the solution of stochastic ordinary differential equations via small delays, Stochastics Stochastics Rep., 28 (1989), 293-299.  doi: 10.1080/17442508908833598.

[3]

M. Benabdallah and M. Bourza, Carathéodory approximate solutions for a class of perturbed stochastic differential equations with reflecting boundary, Stoch. Anal. Appl., 37 (2019), 936-954.  doi: 10.1080/07362994.2019.1623049.

[4]

E. BuckwarR. KuskeS.-E. Mohammed and T. Shardlow, Weak convergence of the Euler scheme for stochastic differential delay equations, LMS J. Comput. Math., 11 (2008), 60-99.  doi: 10.1112/S146115700000053X.

[5]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, , McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.

[6]

N. T. Dung and T. C. Son, Lipschitz continuity in the Hurst index of the solutions of fractional stochastic Volterra integro-differential equations, Submitted.

[7]

F. Faizullah, A note on the Carathéodory approximation scheme for stochastic differential equations under G-Brownian motion, Z. Naturforsch., 67a (2012), 699-704.  doi: 10.5560/zna.2012-0079.

[8]

M. Ferrante and C. Rovira, Convergence of delay differential equations driven by fractional Brownian motion, J. Evol. Equ., 10 (2010), 761-783.  doi: 10.1007/s00028-010-0069-8.

[9]

B. Jourdain and A. Kohatsu-Higa, A review of recent results on approximation of solutions of stochastic differential equations, Stochastic Analysis with Financial Applications, 121–144, Progr. Probab., 65, Birkhäuser/Springer Basel AG, Basel, (2011). doi: 10.1007/978-3-0348-0097-6_9.

[10]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.

[11]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[12]

K. Liu, Carathéodory approximate solutions for a class of semilinear stochastic evolution equations with time delays, J. Math. Anal. Appl., 220 (1998), 349-364.  doi: 10.1006/jmaa.1997.5889.

[13]

W. Mao, L. Hu and X. Mao, Approximate solutions for a class of doubly perturbed stochastic differential equations, Adv. Difference Equ., (2018), Paper No. 37, 17 pp. doi: 10.1186/s13662-018-1490-5.

[14]

X. Mao, Approximate solutions for a class of stochastic evolution equations with variable delays, Numer. Funct. Anal. Optim., 12 (1991), 525–533 (1992). doi: 10.1080/01630569108816448.

[15]

X. Mao, Approximate solutions for a class of stochastic evolution equations with variable delays. II, Numer. Funct. Anal. Optim., 15 (1994), 65-76.  doi: 10.1080/01630569408816550.

[16]

X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.

[17]

D. Nualart, The Malliavin Calculus and Related Topics, Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006.

[18]

B. G. Pachpatte, Inequalities for Differential and Integral Equations, Mathematics in Science and Engineering, 197. Academic Press, Inc., San Diego, CA, 1998.

[19]

J. Turo, Carathéodory approximation solutions to a class of stochastic functional-differential equations, Appl. Anal., 61 (1996), 121-128.  doi: 10.1080/00036819608840450.

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