
Previous Article
Invariance and monotonicity for stochastic delay differential equations
 DCDSB Home
 This Issue

Next Article
Preface
A note on the analysis of asymptotic meansquare stability properties for systems of linear stochastic delay differential equations
1.  Johannes Kepler University, Institute for Stochastics, Altenbergerstraße 69, 4040 Linz, Austria, Austria 
References:
[1] 
L. Arnold, "Stochastic Differential Equations: Theory and Applications," WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 1974. Google Scholar 
[2] 
M. Bandyopadhyay and J. Chattopadhyay, Ratiodependent predatorprey model: Effect of environmental fluctuation and stability, Nonlinearity, 18 (2005), 913936. doi: 10.1088/09517715/18/2/022. Google Scholar 
[3] 
E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Mathematics and Computers in Simulation, 45 (1998), 269277. doi: 10.1016/S03784754(97)001067. Google Scholar 
[4] 
J. Boulet, R. Balasubramaniam, A. Daffertshofer and A. Longtin, Stochastic two delaydifferential model of delayed visual feedback effects on postural dynamics, Philosophical Transactions of the Royal Society A, 368 (2010), 423438. Google Scholar 
[5] 
N. Bradul and L. Shaikhet, Stability of the positive point of equilibrium of Nicholson's blowflies equation with stochastic perturbations: Numerical analysis, Discrete Dynamics in Nature and Society, 2007, Article ID 92959, 25 pp. doi: 10.1155/2007/92959. Google Scholar 
[6] 
D. Bratsun, D. Volfson, L. S. Tsimring and J. Hasty, Delayinduced stochastic oscillations in gene regulation, Proc. Natl. Acad. Sci. USA, 102 (2005), 1459314598. Google Scholar 
[7] 
E. Buckwar and T. Sickenberger, A structural analysis of asymptotic meansquare stability for multidimensional linear stochastic differential systems, Applied Numerical Mathematics, 62 (2012), 842859. doi: 10.1016/j.apnum.2012.03.002. Google Scholar 
[8] 
M. Carletti, On the stability properties of a stochastic model for phagebacteria interaction in open marine environment, Mathematical Biosciences, 175 (2002), 117131. doi: 10.1016/S00255564(01)00089X. Google Scholar 
[9] 
M. Carletti, Mean square stability of a stochastic model for bacteriophage infection with time delays, Mathematical Biosciences, 210 (2007), 395414. doi: 10.1016/j.mbs.2007.05.009. Google Scholar 
[10] 
G. Decoa, V. Jirsa, A. R. McIntosh, O. Sporns and R. Kötter, Key role of coupling, delay, and noise in resting brain fluctuations, Proceedings of the National Academy of Sciences, 106 (2009), 1030210307. Google Scholar 
[11] 
K. Engelborghs, T. Luzyanina and G. Samaey, DDEBIFTOOL v. 2.00: A Matlab package for bifurcation analysis of delay differential equations, Techreport TW330, Department of Computer Science, K. U. Leuven, Leuven, Belgium, 2001. Google Scholar 
[12] 
T. Erneux, "Applied Delay Differential Equations," Surveys and Tutorials in the Applied Mathematical Sciences, 3, Springer, New York, 2009. Google Scholar 
[13] 
K. Green and T. Wagenknecht, Pseudospectra and delay differential equations, Journal of Computational and Applied Mathematics, 196 (2006), 567578. doi: 10.1016/j.cam.2005.10.011. Google Scholar 
[14] 
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences, 99, SpringerVerlag, New York, 1993. Google Scholar 
[15] 
R. Khasminskii, "Stochastic Stability of Differential Equations," With contributions by G. N. Milstein and M. B. Nevelson, Completely revised and enlarged second edition, Stochastic Modelling and Applied Probability, 66, Springer, Heidelberg, 2012. doi: 10.1007/9783642232800. Google Scholar 
[16] 
C. Hauptmann, O. Popovych and P. A.Tass, Multisite coordinated delayed feedback for an effective desynchronization of neuronal networks, Stochastics and Dynamics, 5 (2005), 307319. doi: 10.1142/S0219493705001420. Google Scholar 
[17] 
T. Insperger and G. Stépán, "Semidiscretization for Timedelay Systems. Stability and Engineering Applications," Applied Mathematical Sciences, 178, Springer, New York, 2011. doi: 10.1007/9781461403357. Google Scholar 
[18] 
V. B. Kolmanovskii and A. Myshkis, "Introduction to the Theory and Applications of Functional Differential Equations," Mathematics and its Applications, 463, Kluwer Academic Publishers, Dordrecht, 1999. Google Scholar 
[19] 
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. Google Scholar 
[20] 
H. Lütkepohl, "Handbook of Matrices," John Wiley & Sons, Ltd., Chichester, 1996. Google Scholar 
[21] 
X. Mao, "Stochastic Differential Equations And Applications," Second edition, Horwood Publishing Limited, Chichester, 2008. Google Scholar 
[22] 
S. E. A. Mohammed, "Stochastic Functional Differential Equations," Pitman, Boston, 1984. Google Scholar 
[23] 
L. G. Morelli, S. Ares, L. Herrgen, C. Schröter, F. Jülicher and A. C. Oates, Delayed coupling theory of vertebrate segmentation, HFSP Journal, 3 (2009), 5566. Google Scholar 
[24] 
B. Mukhopadhyay and R. Bhattacharyya, A nonlinear mathematical model of virustumorimmune system interaction: Deterministic and stochastic analysis, Stochastic Analysis and Applications, 27 (2009), 409429. doi: 10.1080/07362990802679067. Google Scholar 
[25] 
M. C. Mackey and I. G. Nechaeva, Solution moment stability in stochastic differential delay equations, Physical Review E (3), 52 (1995), 33663376. doi: 10.1103/PhysRevE.52.3366. Google Scholar 
[26] 
S. Pal, S. Chatterjee, K. pada Das and J. Chattopadhyay, Role of competition in phytoplankton population for the occurrence and control of plankton bloom in the presence of environmental fluctuations, Ecological Modelling, 220 (2009), 96110. Google Scholar 
[27] 
B. Paternoster and L. Shaikhet, Stability of equilibrium points of fractional difference equations with stochastic perturbations, Advances in Difference Equations, 2008, Article ID 718408, 21 pp. Google Scholar 
[28] 
M.K. von Renesse and M. Scheutzow, Existence and uniqueness of solutions of stochastic functional differential equations, Random Operators and Stochastic Equations, 18 (2010), 267284. doi: 10.1515/ROSE.2010.015. Google Scholar 
[29] 
R. R. Sarkar and S. Banerjee, Cancer self remission and tumor stability  a stochastic approach, Mathematical Biosciences, 196 (2005), 6581. doi: 10.1016/j.mbs.2005.04.001. Google Scholar 
[30] 
R. Schlicht and G. Winkler, A delay stochastic process with applications in molecular biology, Journal of Mathematical Biology, 57 (2008), 613648. doi: 10.1007/s002850080178y. Google Scholar 
[31] 
E. Schöll, G. Hiller, P. Hövel and M. Dahlem, Timedelayed feedback in neurosystems, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 367 (2009), 10791096. doi: 10.1098/rsta.2008.0258. Google Scholar 
[32] 
L. Shaikhet, Equations for determining the moments of solutions of linear stochastic differential equations with aftereffect, in "Theory of Stochastic Processes," No. 6 (Russian), 136, "Naukova Dumka," Kiev, (1978), 120123. Google Scholar 
[33] 
L. Shaikhet, Stability of predatorprey model with aftereffect by stochastic perturbations, Stability and Control: Theory and Application, 1 (1998), 313. Google Scholar 
[34] 
L. Shaikhet, Stability of a positive point of equilibrium of one nonlinear system with aftereffect and stochastic perturbations, Dynamic Systems and Applications, 17 (2008), 235253. Google Scholar 
[35] 
L. Shaikhet, "Lyapunov Functionals and Stability of Stochastic Difference Equations," Springer, New York, 2011. Google Scholar 
[36] 
H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences," Texts in Applied Mathematics, 57, Springer, New York, 2011. doi: 10.1007/9781441976468. Google Scholar 
show all references
References:
[1] 
L. Arnold, "Stochastic Differential Equations: Theory and Applications," WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 1974. Google Scholar 
[2] 
M. Bandyopadhyay and J. Chattopadhyay, Ratiodependent predatorprey model: Effect of environmental fluctuation and stability, Nonlinearity, 18 (2005), 913936. doi: 10.1088/09517715/18/2/022. Google Scholar 
[3] 
E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations, Mathematics and Computers in Simulation, 45 (1998), 269277. doi: 10.1016/S03784754(97)001067. Google Scholar 
[4] 
J. Boulet, R. Balasubramaniam, A. Daffertshofer and A. Longtin, Stochastic two delaydifferential model of delayed visual feedback effects on postural dynamics, Philosophical Transactions of the Royal Society A, 368 (2010), 423438. Google Scholar 
[5] 
N. Bradul and L. Shaikhet, Stability of the positive point of equilibrium of Nicholson's blowflies equation with stochastic perturbations: Numerical analysis, Discrete Dynamics in Nature and Society, 2007, Article ID 92959, 25 pp. doi: 10.1155/2007/92959. Google Scholar 
[6] 
D. Bratsun, D. Volfson, L. S. Tsimring and J. Hasty, Delayinduced stochastic oscillations in gene regulation, Proc. Natl. Acad. Sci. USA, 102 (2005), 1459314598. Google Scholar 
[7] 
E. Buckwar and T. Sickenberger, A structural analysis of asymptotic meansquare stability for multidimensional linear stochastic differential systems, Applied Numerical Mathematics, 62 (2012), 842859. doi: 10.1016/j.apnum.2012.03.002. Google Scholar 
[8] 
M. Carletti, On the stability properties of a stochastic model for phagebacteria interaction in open marine environment, Mathematical Biosciences, 175 (2002), 117131. doi: 10.1016/S00255564(01)00089X. Google Scholar 
[9] 
M. Carletti, Mean square stability of a stochastic model for bacteriophage infection with time delays, Mathematical Biosciences, 210 (2007), 395414. doi: 10.1016/j.mbs.2007.05.009. Google Scholar 
[10] 
G. Decoa, V. Jirsa, A. R. McIntosh, O. Sporns and R. Kötter, Key role of coupling, delay, and noise in resting brain fluctuations, Proceedings of the National Academy of Sciences, 106 (2009), 1030210307. Google Scholar 
[11] 
K. Engelborghs, T. Luzyanina and G. Samaey, DDEBIFTOOL v. 2.00: A Matlab package for bifurcation analysis of delay differential equations, Techreport TW330, Department of Computer Science, K. U. Leuven, Leuven, Belgium, 2001. Google Scholar 
[12] 
T. Erneux, "Applied Delay Differential Equations," Surveys and Tutorials in the Applied Mathematical Sciences, 3, Springer, New York, 2009. Google Scholar 
[13] 
K. Green and T. Wagenknecht, Pseudospectra and delay differential equations, Journal of Computational and Applied Mathematics, 196 (2006), 567578. doi: 10.1016/j.cam.2005.10.011. Google Scholar 
[14] 
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences, 99, SpringerVerlag, New York, 1993. Google Scholar 
[15] 
R. Khasminskii, "Stochastic Stability of Differential Equations," With contributions by G. N. Milstein and M. B. Nevelson, Completely revised and enlarged second edition, Stochastic Modelling and Applied Probability, 66, Springer, Heidelberg, 2012. doi: 10.1007/9783642232800. Google Scholar 
[16] 
C. Hauptmann, O. Popovych and P. A.Tass, Multisite coordinated delayed feedback for an effective desynchronization of neuronal networks, Stochastics and Dynamics, 5 (2005), 307319. doi: 10.1142/S0219493705001420. Google Scholar 
[17] 
T. Insperger and G. Stépán, "Semidiscretization for Timedelay Systems. Stability and Engineering Applications," Applied Mathematical Sciences, 178, Springer, New York, 2011. doi: 10.1007/9781461403357. Google Scholar 
[18] 
V. B. Kolmanovskii and A. Myshkis, "Introduction to the Theory and Applications of Functional Differential Equations," Mathematics and its Applications, 463, Kluwer Academic Publishers, Dordrecht, 1999. Google Scholar 
[19] 
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. Google Scholar 
[20] 
H. Lütkepohl, "Handbook of Matrices," John Wiley & Sons, Ltd., Chichester, 1996. Google Scholar 
[21] 
X. Mao, "Stochastic Differential Equations And Applications," Second edition, Horwood Publishing Limited, Chichester, 2008. Google Scholar 
[22] 
S. E. A. Mohammed, "Stochastic Functional Differential Equations," Pitman, Boston, 1984. Google Scholar 
[23] 
L. G. Morelli, S. Ares, L. Herrgen, C. Schröter, F. Jülicher and A. C. Oates, Delayed coupling theory of vertebrate segmentation, HFSP Journal, 3 (2009), 5566. Google Scholar 
[24] 
B. Mukhopadhyay and R. Bhattacharyya, A nonlinear mathematical model of virustumorimmune system interaction: Deterministic and stochastic analysis, Stochastic Analysis and Applications, 27 (2009), 409429. doi: 10.1080/07362990802679067. Google Scholar 
[25] 
M. C. Mackey and I. G. Nechaeva, Solution moment stability in stochastic differential delay equations, Physical Review E (3), 52 (1995), 33663376. doi: 10.1103/PhysRevE.52.3366. Google Scholar 
[26] 
S. Pal, S. Chatterjee, K. pada Das and J. Chattopadhyay, Role of competition in phytoplankton population for the occurrence and control of plankton bloom in the presence of environmental fluctuations, Ecological Modelling, 220 (2009), 96110. Google Scholar 
[27] 
B. Paternoster and L. Shaikhet, Stability of equilibrium points of fractional difference equations with stochastic perturbations, Advances in Difference Equations, 2008, Article ID 718408, 21 pp. Google Scholar 
[28] 
M.K. von Renesse and M. Scheutzow, Existence and uniqueness of solutions of stochastic functional differential equations, Random Operators and Stochastic Equations, 18 (2010), 267284. doi: 10.1515/ROSE.2010.015. Google Scholar 
[29] 
R. R. Sarkar and S. Banerjee, Cancer self remission and tumor stability  a stochastic approach, Mathematical Biosciences, 196 (2005), 6581. doi: 10.1016/j.mbs.2005.04.001. Google Scholar 
[30] 
R. Schlicht and G. Winkler, A delay stochastic process with applications in molecular biology, Journal of Mathematical Biology, 57 (2008), 613648. doi: 10.1007/s002850080178y. Google Scholar 
[31] 
E. Schöll, G. Hiller, P. Hövel and M. Dahlem, Timedelayed feedback in neurosystems, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 367 (2009), 10791096. doi: 10.1098/rsta.2008.0258. Google Scholar 
[32] 
L. Shaikhet, Equations for determining the moments of solutions of linear stochastic differential equations with aftereffect, in "Theory of Stochastic Processes," No. 6 (Russian), 136, "Naukova Dumka," Kiev, (1978), 120123. Google Scholar 
[33] 
L. Shaikhet, Stability of predatorprey model with aftereffect by stochastic perturbations, Stability and Control: Theory and Application, 1 (1998), 313. Google Scholar 
[34] 
L. Shaikhet, Stability of a positive point of equilibrium of one nonlinear system with aftereffect and stochastic perturbations, Dynamic Systems and Applications, 17 (2008), 235253. Google Scholar 
[35] 
L. Shaikhet, "Lyapunov Functionals and Stability of Stochastic Difference Equations," Springer, New York, 2011. Google Scholar 
[36] 
H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences," Texts in Applied Mathematics, 57, Springer, New York, 2011. doi: 10.1007/9781441976468. Google Scholar 
[1] 
Fuke Wu, Peter E. Kloeden. Meansquare random attractors of stochastic delay differential equations with random delay. Discrete & Continuous Dynamical Systems  B, 2013, 18 (6) : 17151734. doi: 10.3934/dcdsb.2013.18.1715 
[2] 
Bixiang Wang. Meansquare random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 14491468. doi: 10.3934/dcds.2020324 
[3] 
Hailong Zhu, Jifeng Chu, Weinian Zhang. Meansquare almost automorphic solutions for stochastic differential equations with hyperbolicity. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 19351953. doi: 10.3934/dcds.2018078 
[4] 
Chuchu Chen, Jialin Hong. Meansquare convergence of numerical approximations for a class of backward stochastic differential equations. Discrete & Continuous Dynamical Systems  B, 2013, 18 (8) : 20512067. doi: 10.3934/dcdsb.2013.18.2051 
[5] 
Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The meansquare dichotomy spectrum and a bifurcation to a meansquare attractor. Discrete & Continuous Dynamical Systems  B, 2015, 20 (3) : 875887. doi: 10.3934/dcdsb.2015.20.875 
[6] 
Kai Du, Jianhui Huang, Zhen Wu. Linear quadratic meanfieldgame of backward stochastic differential systems. Mathematical Control & Related Fields, 2018, 8 (3&4) : 653678. doi: 10.3934/mcrf.2018028 
[7] 
Quan Hai, Shutang Liu. Meansquare delaydistributiondependent exponential synchronization of chaotic neural networks with mixed random timevarying delays and restricted disturbances. Discrete & Continuous Dynamical Systems  B, 2021, 26 (6) : 30973118. doi: 10.3934/dcdsb.2020221 
[8] 
Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems  B, 2019, 24 (10) : 57095736. doi: 10.3934/dcdsb.2019103 
[9] 
Wei Wang, Kai Liu, Xiulian Wang. Sensitivity to small delays of mean square stability for stochastic neutral evolution equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 24032418. doi: 10.3934/cpaa.2020105 
[10] 
Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493505. doi: 10.3934/eect.2015.4.493 
[11] 
Jun Moon. Linearquadratic meanfield type stackelberg differential games for stochastic jumpdiffusion systems. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021026 
[12] 
Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 13611375. doi: 10.3934/cpaa.2011.10.1361 
[13] 
A. Domoshnitsky. About maximum principles for one of the components of solution vector and stability for systems of linear delay differential equations. Conference Publications, 2011, 2011 (Special) : 373380. doi: 10.3934/proc.2011.2011.373 
[14] 
Ziheng Chen, Siqing Gan, Xiaojie Wang. Meansquare approximations of Lévy noise driven SDEs with superlinearly growing diffusion and jump coefficients. Discrete & Continuous Dynamical Systems  B, 2019, 24 (8) : 45134545. doi: 10.3934/dcdsb.2019154 
[15] 
Pablo Pedregal. Fully explicit quasiconvexification of the meansquare deviation of the gradient of the state in optimal design. Electronic Research Announcements, 2001, 7: 7278. 
[16] 
Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479502. doi: 10.3934/jgm.2014.6.479 
[17] 
Yong He. Switching controls for linear stochastic differential systems. Mathematical Control & Related Fields, 2020, 10 (2) : 443454. doi: 10.3934/mcrf.2020005 
[18] 
Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete & Continuous Dynamical Systems  B, 2013, 18 (6) : 16831696. doi: 10.3934/dcdsb.2013.18.1683 
[19] 
Zhen Wang, Xiong Li, Jinzhi Lei. Second moment boundedness of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems  B, 2014, 19 (9) : 29632991. doi: 10.3934/dcdsb.2014.19.2963 
[20] 
Weijun Meng, Jingtao Shi. A linear quadratic stochastic Stackelberg differential game with time delay. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021035 
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]