
Previous Article
$Z^d$ Toeplitz arrays
 DCDS Home
 This Issue

Next Article
On the Euler equation for minimal geodesics on Riemannian manifoldshaving discontinuous metrics
On stochastic stabilization of difference equations
1.  School of Mathematical Sciences, Dublin City University, Dublin, Ireland 
2.  Department of Statistics and Modelling Science, University of Strathclyde, Glasgow, G1 1XH, Scotland, United Kingdom 
3.  Department of Mathematics and Computer Science, The University of the West Indies, Mona, Kingston 7, Jamaica 
$x_{n+1}=x_n(1+a_nf(x_n))$, $n\ge 1$, $x_0=a$.
We show how this equation can be stabilized by adding the random noise term $\sigma_ng(x_n)\xi_{n+1}$ where $\xi_n$ takes the values +1 or 1 each with probability $1/2$. We also prove a theorem on the almost sure asymptotic stability of the solution of a scalar nonlinear stochastic difference equation with bounded coefficients, and show the connection between the noise stabilization of a stochastic differential equation, and a discretization of this equation.
[1] 
Wensheng Yin, Jinde Cao. Almost sure exponential stabilization and suppression by periodically intermittent stochastic perturbation with jumps. Discrete and Continuous Dynamical Systems  B, 2020, 25 (11) : 44934513. doi: 10.3934/dcdsb.2020109 
[2] 
Alexandra Rodkina, Henri Schurz, Leonid Shaikhet. Almost sure stability of some stochastic dynamical systems with memory. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 571593. doi: 10.3934/dcds.2008.21.571 
[3] 
Gregory Berkolaiko, Cónall Kelly, Alexandra Rodkina. Sharp pathwise asymptotic stability criteria for planar systems of linear stochastic difference equations. Conference Publications, 2011, 2011 (Special) : 163173. doi: 10.3934/proc.2011.2011.163 
[4] 
Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 10051023. doi: 10.3934/dcds.2009.24.1005 
[5] 
Minghui Song, Liangjian Hu, Xuerong Mao, Liguo Zhang. Khasminskiitype theorems for stochastic functional differential equations. Discrete and Continuous Dynamical Systems  B, 2013, 18 (6) : 16971714. doi: 10.3934/dcdsb.2013.18.1697 
[6] 
Hailong Zhu, Jifeng Chu, Weinian Zhang. Meansquare almost automorphic solutions for stochastic differential equations with hyperbolicity. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 19351953. doi: 10.3934/dcds.2018078 
[7] 
Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete and Continuous Dynamical Systems  B, 2019, 24 (11) : 59275944. doi: 10.3934/dcdsb.2019113 
[8] 
Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete and Continuous Dynamical Systems  B, 2008, 9 (1) : 4764. doi: 10.3934/dcdsb.2008.9.47 
[9] 
Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the splitstep theta method for stochastic differential equations with piecewise continuous arguments. Discrete and Continuous Dynamical Systems  B, 2019, 24 (2) : 695717. doi: 10.3934/dcdsb.2018203 
[10] 
Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 59295949. doi: 10.3934/dcds.2016060 
[11] 
Wei Mao, Yanan Jiang, Liangjian Hu, Xuerong Mao. Stabilization by intermittent control for hybrid stochastic differential delay equations. Discrete and Continuous Dynamical Systems  B, 2022, 27 (1) : 569581. doi: 10.3934/dcdsb.2021055 
[12] 
Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic meansquare stability properties for systems of linear stochastic delay differential equations. Discrete and Continuous Dynamical Systems  B, 2013, 18 (6) : 15211531. doi: 10.3934/dcdsb.2013.18.1521 
[13] 
Wei Mao, Liangjian Hu, Xuerong Mao. Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the EulerMaruyama approximation. Discrete and Continuous Dynamical Systems  B, 2019, 24 (2) : 587613. doi: 10.3934/dcdsb.2018198 
[14] 
Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete and Continuous Dynamical Systems  B, 2015, 20 (1) : 2338. doi: 10.3934/dcdsb.2015.20.23 
[15] 
Tian Zhang, Huabin Chen, Chenggui Yuan, Tomás Caraballo. On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations. Discrete and Continuous Dynamical Systems  B, 2019, 24 (10) : 53555375. doi: 10.3934/dcdsb.2019062 
[16] 
Alexandra Rodkina, Henri Schurz. On positivity and boundedness of solutions of nonlinear stochastic difference equations. Conference Publications, 2009, 2009 (Special) : 640649. doi: 10.3934/proc.2009.2009.640 
[17] 
Yoshihiro Hamaya. Stability properties and existence of almost periodic solutions of volterra difference equations. Conference Publications, 2009, 2009 (Special) : 315321. doi: 10.3934/proc.2009.2009.315 
[18] 
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 
[19] 
Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 295313. doi: 10.3934/dcds.2007.18.295 
[20] 
Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete and Continuous Dynamical Systems  B, 2017, 22 (7) : 29232938. doi: 10.3934/dcdsb.2017157 
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]