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Instability of unidirectional flows for the 2D α-Euler equations
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 |
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 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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