On stochastic stabilization of difference equations

@article{Appleby2006OnSS,
  title={On stochastic stabilization of difference equations},
  author={John A. D. Appleby and Xuerong Mao and Alexandra Rodkina},
  journal={Discrete and Continuous Dynamical Systems},
  year={2006},
  volume={15},
  pages={843-857}
}
We consider unstable scalar deterministic difference equation $x_{n+1}=x_n(1+a_nf(x_n))$, $n\ge 1$, $x_0=a$. 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… 
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References

SHOWING 1-10 OF 15 REFERENCES
On boundedness and stability of solutions of nonlinear difference equation with nonmartingale type noise
Consider a stochastic difference equation with the Volterra type nonlinear main term G and Volterra type noise Functions Gf, σ supposed to be random, x;i is a martingale—difference. In general so Eq,
Stochastic stabilisation of functional differential equations
ON GLOBAL ASYMPTOTIC STABILITY OF SOLUTIONS OF SOME IN-ARITHMETIC-MEAN-SENSE MONOTONE STOCHASTIC DIFFERENCE EQUATIONS IN IR
Global almost sure asymptotic stability of the trivial solution of some nonlinear stochastic difference equations with in-the-arithmetic-meansense monotone drift part and diffusive part driven by
Global asymptotic stability of solutions of cubic stochastic difference equations
Global almost sure asymptotic stability of solutions of some nonlinear stochastic difference equations with cubic-type main part in their drift and diffusive part driven by square-integrable
On asymptotic behaviour of solutions of stochastic difference equations with volterra type main term
The stochastic difference equations with Volterra type linear and nonlinear main term are considered in the paper.Conditions on a.s. boundedness of the solutions,asymptotic stability,exponential
Stochastic self-stabilization
Given a nonlinear Ito equation suppose that the equation is almost surely exponentially stable provided the noise intensity u is large enough. We show that if u is replaced by then the equation
Brownian Motion and Stochastic Calculus
TLDR
This chapter discusses Brownian motion, which is concerned with continuous, Square-Integrable Martingales, and the Stochastic Integration, which deals with the integration of continuous, local martingales into Markov processes.
Stabilization of Volterra equations by noise
The paper studies the stability of an autonomous convolution Ito-Volterra equation where the linear diffusion term depends on the current value of the state only, and the memory of the past fades
Stochastic stabilization of differential systems with general decay rate
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