
Previous Article
Meansquare random attractors of stochastic delay differential equations with random delay
 DCDSB Home
 This Issue

Next Article
Exponential growth rate for a singular linear stochastic delay differential equation
Khasminskiitype theorems for stochastic functional differential equations
1.  Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China 
2.  Department of Applied Mathematics, Donghua Univerisity, Shanghai 201600, China 
3.  Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH 
4.  School of Electronic Information and Control Engineering, Beijing University of Technology, Beijing 100124, China 
References:
[1] 
A. Bahar and X. Mao, Stochastic delay LotkaVolterra model, J. Math. Anal. Appl., 292 (2004), 364380. doi: 10.1016/j.jmaa.2003.12.004. 
[2] 
A. Bahar and X. Mao, Stochastic delay population dynamics, International J. Pure and Applied Math., 11 (2004), 377400. 
[3] 
T. C. Gard, "Introduction to Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, Inc., New York, 1988. 
[4] 
R. Z. Has'minskiĭ, "Stochastic Stability of Differential Equations," Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den RijnGermantown, Md., 1980. 
[5] 
V. Kolmanovskiĭ and A. Myshkis, "Applied Theory of FunctionalDifferential Equations," Mathematics and its Applications (Soviet Series), 85, Kluwer Academic Publishers Group, Dordrecht, 1992. 
[6] 
G. S. Ladde and V. Lakshmikantham, "Random Differential Inequalities," Mathematics in Science and Engineering, 150, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1980. 
[7] 
M. Loève, "Probability Theory," Third edition, D. Van Nostrand Company, Inc., Princeton, N. J.Toronto, Ont.London, 1963. 
[8] 
Q. Luo, X. Mao and Y. Shen, Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations, Automatica J. IFAC, 47 (2011), 20752081. doi: 10.1016/j.automatica.2011.06.014. 
[9] 
X. Mao, "Stability of Stochastic Differential Equations with Respect to Semimartingales," Pitman Research Notes in Mathematics Series, 251, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. 
[10] 
X. Mao, "Exponential Stability of Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 182, Marcel Dekker, Inc., New York, 1994. 
[11] 
X. Mao, "Stochastic Differential Equations and Applications," Second edition, Horwood Publishing Limited, Chichester, 2008. 
[12] 
X. Mao, A note on the LaSalletype theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2002), 125142. doi: 10.1006/jmaa.2001.7803. 
[13] 
X. Mao and M. J. Rassias, Khasminskiitype theorems for stochastic differential delay equations, J. Sto. Anal. Appl., 23 (2005), 10451069. doi: 10.1080/07362990500118637. 
[14] 
X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching," Imperial College Press, London, 2006. 
[15] 
X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations in population dynamics, J. Math. Anal. Appl., 304 (2005), 296320. doi: 10.1016/j.jmaa.2004.09.027. 
[16] 
S. E. A. Mohammed, "Stochastic Functional Differential Equations," Research Notes in Mathematics, 99, Pitman (Advanced Publishing Program), Boston, MA, 1984. 
show all references
References:
[1] 
A. Bahar and X. Mao, Stochastic delay LotkaVolterra model, J. Math. Anal. Appl., 292 (2004), 364380. doi: 10.1016/j.jmaa.2003.12.004. 
[2] 
A. Bahar and X. Mao, Stochastic delay population dynamics, International J. Pure and Applied Math., 11 (2004), 377400. 
[3] 
T. C. Gard, "Introduction to Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, Inc., New York, 1988. 
[4] 
R. Z. Has'minskiĭ, "Stochastic Stability of Differential Equations," Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den RijnGermantown, Md., 1980. 
[5] 
V. Kolmanovskiĭ and A. Myshkis, "Applied Theory of FunctionalDifferential Equations," Mathematics and its Applications (Soviet Series), 85, Kluwer Academic Publishers Group, Dordrecht, 1992. 
[6] 
G. S. Ladde and V. Lakshmikantham, "Random Differential Inequalities," Mathematics in Science and Engineering, 150, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1980. 
[7] 
M. Loève, "Probability Theory," Third edition, D. Van Nostrand Company, Inc., Princeton, N. J.Toronto, Ont.London, 1963. 
[8] 
Q. Luo, X. Mao and Y. Shen, Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations, Automatica J. IFAC, 47 (2011), 20752081. doi: 10.1016/j.automatica.2011.06.014. 
[9] 
X. Mao, "Stability of Stochastic Differential Equations with Respect to Semimartingales," Pitman Research Notes in Mathematics Series, 251, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. 
[10] 
X. Mao, "Exponential Stability of Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 182, Marcel Dekker, Inc., New York, 1994. 
[11] 
X. Mao, "Stochastic Differential Equations and Applications," Second edition, Horwood Publishing Limited, Chichester, 2008. 
[12] 
X. Mao, A note on the LaSalletype theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2002), 125142. doi: 10.1006/jmaa.2001.7803. 
[13] 
X. Mao and M. J. Rassias, Khasminskiitype theorems for stochastic differential delay equations, J. Sto. Anal. Appl., 23 (2005), 10451069. doi: 10.1080/07362990500118637. 
[14] 
X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching," Imperial College Press, London, 2006. 
[15] 
X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations in population dynamics, J. Math. Anal. Appl., 304 (2005), 296320. doi: 10.1016/j.jmaa.2004.09.027. 
[16] 
S. E. A. Mohammed, "Stochastic Functional Differential Equations," Research Notes in Mathematics, 99, Pitman (Advanced Publishing Program), Boston, MA, 1984. 
[1] 
Defei Zhang, Ping He. Functional solution about stochastic differential equation driven by $G$Brownian motion. Discrete and Continuous Dynamical Systems  B, 2015, 20 (1) : 281293. doi: 10.3934/dcdsb.2015.20.281 
[2] 
Brahim Boufoussi, Soufiane Mouchtabih. Controllability of neutral stochastic functional integrodifferential equations driven by fractional brownian motion with Hurst parameter lesser than $ 1/2 $. Evolution Equations and Control Theory, 2021, 10 (4) : 921935. doi: 10.3934/eect.2020096 
[3] 
Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by onedimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237248. doi: 10.3934/mbe.2017015 
[4] 
María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems  B, 2010, 14 (2) : 473493. doi: 10.3934/dcdsb.2010.14.473 
[5] 
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of pth mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems  B, 2020, 25 (3) : 11411158. doi: 10.3934/dcdsb.2019213 
[6] 
Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$Brownian motion. Discrete and Continuous Dynamical Systems  B, 2015, 20 (7) : 21572169. doi: 10.3934/dcdsb.2015.20.2157 
[7] 
Shaokuan Chen, Shanjian Tang. Semilinear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401434. doi: 10.3934/mcrf.2015.5.401 
[8] 
Ahmed Boudaoui, Tomás Caraballo, Abdelghani Ouahab. Stochastic differential equations with noninstantaneous impulses driven by a fractional Brownian motion. Discrete and Continuous Dynamical Systems  B, 2017, 22 (7) : 25212541. doi: 10.3934/dcdsb.2017084 
[9] 
Litan Yan, Xiuwei Yin. Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion. Discrete and Continuous Dynamical Systems  B, 2019, 24 (2) : 615635. doi: 10.3934/dcdsb.2018199 
[10] 
Fuke Wu, Xuerong Mao, Peter E. Kloeden. Discrete Razumikhintype technique and stability of the EulerMaruyama method to stochastic functional differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 885903. doi: 10.3934/dcds.2013.33.885 
[11] 
Fuke Wu, Shigeng Hu. The LaSalletype theorem for neutral stochastic functional differential equations with infinite delay. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 10651094. doi: 10.3934/dcds.2012.32.1065 
[12] 
Ya Wang, Fuke Wu, Xuerong Mao, Enwen Zhu. Advances in the LaSalletype theorems for stochastic functional differential equations with infinite delay. Discrete and Continuous Dynamical Systems  B, 2020, 25 (1) : 287300. doi: 10.3934/dcdsb.2019182 
[13] 
Haiyan Zhang. A necessary condition for meanfield type stochastic differential equations with correlated state and observation noises. Journal of Industrial and Management Optimization, 2016, 12 (4) : 12871301. doi: 10.3934/jimo.2016.12.1287 
[14] 
Yongqiang Suo, Chenggui Yuan. Large deviations for neutral stochastic functional differential equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 23692384. doi: 10.3934/cpaa.2020103 
[15] 
Leonid Shaikhet. Stability of delay differential equations with fading stochastic perturbations of the type of white noise and poisson's jumps. Discrete and Continuous Dynamical Systems  B, 2020, 25 (9) : 36513657. doi: 10.3934/dcdsb.2020077 
[16] 
Jun Zhou, Jun Shen. Positive solutions of iterative functional differential equations and application to mixedtype functional differential equations. Discrete and Continuous Dynamical Systems  B, 2022, 27 (7) : 36053624. doi: 10.3934/dcdsb.2021198 
[17] 
Guolian Wang, Boling Guo. Stochastic Kortewegde Vries equation driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 52555272. doi: 10.3934/dcds.2015.35.5255 
[18] 
Yong Xu, Rong Guo, Di Liu, Huiqing Zhang, Jinqiao Duan. Stochastic averaging principle for dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems  B, 2014, 19 (4) : 11971212. doi: 10.3934/dcdsb.2014.19.1197 
[19] 
Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slowfast dynamical systems with fractional Brownian motion. Discrete and Continuous Dynamical Systems  B, 2015, 20 (7) : 22572267. doi: 10.3934/dcdsb.2015.20.2257 
[20] 
Kai Liu. On regularity of stochastic convolutions of functional linear differential equations with memory. Discrete and Continuous Dynamical Systems  B, 2020, 25 (4) : 12791298. doi: 10.3934/dcdsb.2019220 
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]