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Instability of unidirectional flows for the 2D αEuler equations
Advances in the truncated Euler–Maruyama method for stochastic differential delay equations
1.  Key Laboratory of Advanced Perception and Intelligent Control of Highend 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 
Guo et al. [
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 onestep methods for stochastic delay differential equations, LMS J. Comput. Math., 3 (2000), 315335. 
[3] 
C. T. H. Baker and E. Buckwar, Exponential stability in $p$th mean of solutions, and of convergent Eulertype solutions, of stochastic delay differential equations, J. Comput. Appl. Math.%Discrete Continuous Dynam. Systems, 184 (2005), 404427. 
[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), 863880. 
[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), 155170. 
[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), 232257. 
[8] 
Q. Guo, X. Mao and R. Yue, The truncated EulerMaruyama method for stochastic differential delay equations, Numerical Algorithms, 78 (2018), 599624. 
[9] 
D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the meanreverting square root process, J. Comput. Finance, 8 (2005), 3562. 
[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), 189205. 
[15] 
G. S. Ladde and V. Lakshmikantham, Ramdom Differential Inequalities, Academic Press, 1980. 
[16] 
X. Mao, A note on the LaSalletype theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2000), 125142. 
[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 Khasminskiitype conditions, Appl. Math. Comput., 217 (2011), 55125524. 
[19] 
X. Mao, The truncated EulerMaruyama method for stochastic differential equations, J. Comput. Appl. Math., 290 (2015), 370384. 
[20] 
X. Mao and M. J. Rassias, Khasminskiitype theorems for stochastic differential delay equations, J. Sto. Anal. Appl., 23 (2015), 10451069. 
[21] 
X. Mao and S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, J. Comput. Appl. Math., 151 (2003), 215227. 
[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), 185. 
[25] 
F. Wu and X. Mao, Numerical solutions of neutral stochastic functional differential equations, SIAM J. Numer. Anal., 46 (2008), 18211841. 
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 onestep methods for stochastic delay differential equations, LMS J. Comput. Math., 3 (2000), 315335. 
[3] 
C. T. H. Baker and E. Buckwar, Exponential stability in $p$th mean of solutions, and of convergent Eulertype solutions, of stochastic delay differential equations, J. Comput. Appl. Math.%Discrete Continuous Dynam. Systems, 184 (2005), 404427. 
[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), 863880. 
[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), 155170. 
[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), 232257. 
[8] 
Q. Guo, X. Mao and R. Yue, The truncated EulerMaruyama method for stochastic differential delay equations, Numerical Algorithms, 78 (2018), 599624. 
[9] 
D. J. Higham and X. Mao, Convergence of Monte Carlo simulations involving the meanreverting square root process, J. Comput. Finance, 8 (2005), 3562. 
[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), 189205. 
[15] 
G. S. Ladde and V. Lakshmikantham, Ramdom Differential Inequalities, Academic Press, 1980. 
[16] 
X. Mao, A note on the LaSalletype theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2000), 125142. 
[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 Khasminskiitype conditions, Appl. Math. Comput., 217 (2011), 55125524. 
[19] 
X. Mao, The truncated EulerMaruyama method for stochastic differential equations, J. Comput. Appl. Math., 290 (2015), 370384. 
[20] 
X. Mao and M. J. Rassias, Khasminskiitype theorems for stochastic differential delay equations, J. Sto. Anal. Appl., 23 (2015), 10451069. 
[21] 
X. Mao and S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, J. Comput. Appl. Math., 151 (2003), 215227. 
[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), 185. 
[25] 
F. Wu and X. Mao, Numerical solutions of neutral stochastic functional differential equations, SIAM J. Numer. Anal., 46 (2008), 18211841. 
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