Advances in the truncated Euler–Maruyama method for stochastic differential delay equations

Weiyin Fei; Liangjian Hu; Xuerong Mao; Dengfeng Xia

doi:10.3934/cpaa.2020092

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April 2020, 19(4): 2081-2100. doi: 10.3934/cpaa.2020092

Advances in the truncated Euler–Maruyama method for stochastic differential delay equations

Weiyin Fei ^1, , Liangjian Hu ^2, , Xuerong Mao ^3, and Dengfeng Xia ^4,

1.	Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education, School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, Anhui 241000, China
2.	Department of Applied Mathematics, Donghua Univerisity, Shanghai 201620, China
3.	Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, U.K
4.	School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, Anhui 241000, China

Dedicated to Prof. Dr. Tomás Caraballo's 60th birthday

Received February 2019 Revised July 2019 Published January 2020

Fund Project: This research was partially supported by the Royal Society (WM160014, Royal Society Wolfson Research Merit Award), the Royal Society and the Newton Fund (NA160317, Royal SocietyNewton Advanced Fellowship), the EPSRC (EP/K503174/1), the National Natural Science Foundation of China (61673103, 71571001), the Natural Science Foundation of Shanghai (17ZR1401300) and the Ministry of Education (MOE) of China (MS2014DHDX020).

Guo et al. [8] are the first to study the strong convergence of the explicit numerical method for the highly nonlinear stochastic differential delay equations (SDDEs) under the generalised Khasminskii-type condition. The method used there is the truncated Euler–Maruyama (EM) method. In this paper we will point out that a main condition imposed in [8] is somehow restrictive in the sense that the condition could force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is then to establish the convergence rate without this restriction.

Keywords: Brownian motion, stochastic differential delay equation, Itô's formula, truncated Euler–Maruyama, Khasminskii-type condition.

Mathematics Subject Classification: Primary: 60H10, 60J65.

Citation: Weiyin Fei, Liangjian Hu, Xuerong Mao, Dengfeng Xia. Advances in the truncated Euler–Maruyama method for stochastic differential delay equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2081-2100. doi: 10.3934/cpaa.2020092

References:

[1]	L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley and Sons, 1957.
[2]	C. T. H. Baker and E. Buckwar, Numerical analysis of explicit one-step methods for stochastic delay differential equations, LMS J. Comput. Math., 3 (2000), 315-335.
[3]	C. T. H. Baker and E. Buckwar, Exponential stability in $p$-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations, J. Comput. Appl. Math.%Discrete Continuous Dynam. Systems, 184 (2005), 404-427.
[4]	T. Caraballo, P. E. Kloeden and J. Real, Discretization of asymptotically stable stationary solutions of delay differential equations with a random stationary delay, J. Dyn. & Diff. Eqns., 18 (2006), 863-880.
[5]	T. Caraballo, K. Liu and X. Mao, On stabilization of partial differential equations by noise, Nagoya Mathematical Journal%Discrete Continuous Dynam. Systems, 161 (2001), 155-170.
[6]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univeristy Press, 1992.
[7]	S. Deng, W. Fei, W. Liu and X. Mao, The truncated EM method for stochastic differential equations with Poisson jumps, J. Comput. Appl. Math., 355 (2019), 232-257.
[8]	Q. Guo, X. Mao and R. Yue, The truncated Euler--Maruyama method for stochastic differential delay equations, Numerical Algorithms, 78 (2018), 599-624.
[9]	D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process, J. Comput. Finance, 8 (2005), 35-62.
[10]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1988.
[11]	R. Z. Khasminskii, Stochastic Stability of Differential Equations, Alphen: Sijtjoff and Noordhoff (translation of the Russian edition, Moscow, Nauka 1969), 1980.
[12]	P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, SpringerVerlag, New York, 1992.
[13]	V. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, 1992.
[14]	U. Küchler and E. Platen, Strong discrete time approximation of stochastic differential equations with time delay, Math. Comput. Simul., 54 (2000), 189-205.
[15]	G. S. Ladde and V. Lakshmikantham, Ramdom Differential Inequalities, Academic Press, 1980.
[16]	X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2000), 125-142.
[17]	X. Mao, Stochastic Differential Equations and Applications, 2nd Edition, Horwood Publishing, Chichester, 2007.
[18]	X. Mao, Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions, Appl. Math. Comput., 217 (2011), 5512-5524.
[19]	X. Mao, The truncated Euler--Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 290 (2015), 370-384.
[20]	X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, J. Sto. Anal. Appl., 23 (2015), 1045-1069.
[21]	X. Mao and S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, J. Comput. Appl. Math., 151 (2003), 215-227.
[22]	G. N. Milstein, Numerical Integration of Stochastic Differential Equations, Kluwer Academic Publishers, Dodrecht, 1995.
[23]	S–E. A. Mohammed, Stochastic Functional Differential Equations, Longman Scientific and Technical, 1985.
[24]	H. Schurz, Applications of numerical methods and its analysis for systems of stochastic differential equations, Bull. Kerala Math. Assoc., 4 (2007), 1-85.
[25]	F. Wu and X. Mao, Numerical solutions of neutral stochastic functional differential equations, SIAM J. Numer. Anal., 46 (2008), 1821-1841.

show all references

References:

[1]	L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley and Sons, 1957.
[2]	C. T. H. Baker and E. Buckwar, Numerical analysis of explicit one-step methods for stochastic delay differential equations, LMS J. Comput. Math., 3 (2000), 315-335.
[3]	C. T. H. Baker and E. Buckwar, Exponential stability in $p$-th mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations, J. Comput. Appl. Math.%Discrete Continuous Dynam. Systems, 184 (2005), 404-427.
[4]	T. Caraballo, P. E. Kloeden and J. Real, Discretization of asymptotically stable stationary solutions of delay differential equations with a random stationary delay, J. Dyn. & Diff. Eqns., 18 (2006), 863-880.
[5]	T. Caraballo, K. Liu and X. Mao, On stabilization of partial differential equations by noise, Nagoya Mathematical Journal%Discrete Continuous Dynam. Systems, 161 (2001), 155-170.
[6]	G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univeristy Press, 1992.
[7]	S. Deng, W. Fei, W. Liu and X. Mao, The truncated EM method for stochastic differential equations with Poisson jumps, J. Comput. Appl. Math., 355 (2019), 232-257.
[8]	Q. Guo, X. Mao and R. Yue, The truncated Euler--Maruyama method for stochastic differential delay equations, Numerical Algorithms, 78 (2018), 599-624.
[9]	D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the mean-reverting square root process, J. Comput. Finance, 8 (2005), 35-62.
[10]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1988.
[11]	R. Z. Khasminskii, Stochastic Stability of Differential Equations, Alphen: Sijtjoff and Noordhoff (translation of the Russian edition, Moscow, Nauka 1969), 1980.
[12]	P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, SpringerVerlag, New York, 1992.
[13]	V. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, 1992.
[14]	U. Küchler and E. Platen, Strong discrete time approximation of stochastic differential equations with time delay, Math. Comput. Simul., 54 (2000), 189-205.
[15]	G. S. Ladde and V. Lakshmikantham, Ramdom Differential Inequalities, Academic Press, 1980.
[16]	X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2000), 125-142.
[17]	X. Mao, Stochastic Differential Equations and Applications, 2nd Edition, Horwood Publishing, Chichester, 2007.
[18]	X. Mao, Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions, Appl. Math. Comput., 217 (2011), 5512-5524.
[19]	X. Mao, The truncated Euler--Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 290 (2015), 370-384.
[20]	X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, J. Sto. Anal. Appl., 23 (2015), 1045-1069.
[21]	X. Mao and S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, J. Comput. Appl. Math., 151 (2003), 215-227.
[22]	G. N. Milstein, Numerical Integration of Stochastic Differential Equations, Kluwer Academic Publishers, Dodrecht, 1995.
[23]	S–E. A. Mohammed, Stochastic Functional Differential Equations, Longman Scientific and Technical, 1985.
[24]	H. Schurz, Applications of numerical methods and its analysis for systems of stochastic differential equations, Bull. Kerala Math. Assoc., 4 (2007), 1-85.
[25]	F. Wu and X. Mao, Numerical solutions of neutral stochastic functional differential equations, SIAM J. Numer. Anal., 46 (2008), 1821-1841.

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