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doi: 10.3934/dcdsb.2021249
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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 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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

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

[18]

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

[19]

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

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