# American Institute of Mathematical Sciences

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

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

 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.

Citation: Weiyin Fei, Liangjian Hu, Xuerong Mao, Dengfeng Xia. Advances in the truncated Euler–Maruyama method for stochastic differential delay equations. Communications on Pure & 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. Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univeristy Press, 1992. Google Scholar [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.   Google Scholar [8] Q. Guo, X. Mao and R. Yue, The truncated Euler--Maruyama method for stochastic differential delay equations, Numerical Algorithms, 78 (2018), 599-624.   Google Scholar [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.   Google Scholar [10] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1988. Google Scholar [11] R. Z. Khasminskii, Stochastic Stability of Differential Equations, Alphen: Sijtjoff and Noordhoff (translation of the Russian edition, Moscow, Nauka 1969), 1980. Google Scholar [12] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, SpringerVerlag, New York, 1992. Google Scholar [13] V. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, 1992. Google Scholar [14] U. Küchler and E. Platen, Strong discrete time approximation of stochastic differential equations with time delay, Math. Comput. Simul., 54 (2000), 189-205.   Google Scholar [15] G. S. Ladde and V. Lakshmikantham, Ramdom Differential Inequalities, Academic Press, 1980. Google Scholar [16] X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2000), 125-142.   Google Scholar [17] X. Mao, Stochastic Differential Equations and Applications, 2nd Edition, Horwood Publishing, Chichester, 2007. Google Scholar [18] X. Mao, Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions, Appl. Math. Comput., 217 (2011), 5512-5524.   Google Scholar [19] X. Mao, The truncated Euler--Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 290 (2015), 370-384.   Google Scholar [20] X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, J. Sto. Anal. Appl., 23 (2015), 1045-1069.   Google Scholar [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.   Google Scholar [22] G. N. Milstein, Numerical Integration of Stochastic Differential Equations, Kluwer Academic Publishers, Dodrecht, 1995. Google Scholar [23] S–E. A. Mohammed, Stochastic Functional Differential Equations, Longman Scientific and Technical, 1985. Google Scholar [24] H. Schurz, Applications of numerical methods and its analysis for systems of stochastic differential equations, Bull. Kerala Math. Assoc., 4 (2007), 1-85.   Google Scholar [25] F. Wu and X. Mao, Numerical solutions of neutral stochastic functional differential equations, SIAM J. Numer. Anal., 46 (2008), 1821-1841.   Google Scholar

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##### References:
 [1] L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley and Sons, 1957. Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [6] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univeristy Press, 1992. Google Scholar [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.   Google Scholar [8] Q. Guo, X. Mao and R. Yue, The truncated Euler--Maruyama method for stochastic differential delay equations, Numerical Algorithms, 78 (2018), 599-624.   Google Scholar [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.   Google Scholar [10] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1988. Google Scholar [11] R. Z. Khasminskii, Stochastic Stability of Differential Equations, Alphen: Sijtjoff and Noordhoff (translation of the Russian edition, Moscow, Nauka 1969), 1980. Google Scholar [12] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, SpringerVerlag, New York, 1992. Google Scholar [13] V. Kolmanovskii and A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer Academic Publishers, 1992. Google Scholar [14] U. Küchler and E. Platen, Strong discrete time approximation of stochastic differential equations with time delay, Math. Comput. Simul., 54 (2000), 189-205.   Google Scholar [15] G. S. Ladde and V. Lakshmikantham, Ramdom Differential Inequalities, Academic Press, 1980. Google Scholar [16] X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2000), 125-142.   Google Scholar [17] X. Mao, Stochastic Differential Equations and Applications, 2nd Edition, Horwood Publishing, Chichester, 2007. Google Scholar [18] X. Mao, Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions, Appl. Math. Comput., 217 (2011), 5512-5524.   Google Scholar [19] X. Mao, The truncated Euler--Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 290 (2015), 370-384.   Google Scholar [20] X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, J. Sto. Anal. Appl., 23 (2015), 1045-1069.   Google Scholar [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.   Google Scholar [22] G. N. Milstein, Numerical Integration of Stochastic Differential Equations, Kluwer Academic Publishers, Dodrecht, 1995. Google Scholar [23] S–E. A. Mohammed, Stochastic Functional Differential Equations, Longman Scientific and Technical, 1985. Google Scholar [24] H. Schurz, Applications of numerical methods and its analysis for systems of stochastic differential equations, Bull. Kerala Math. Assoc., 4 (2007), 1-85.   Google Scholar [25] F. Wu and X. Mao, Numerical solutions of neutral stochastic functional differential equations, SIAM J. Numer. Anal., 46 (2008), 1821-1841.   Google Scholar
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