On stochastic stabilization of difference equations
On stochastic stabilization of difference equations
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DOI：
10.3934/dcds.2006.15.843
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发表时间：
2006-04
影响因子：
1.1
通讯作者：
J. Appleby;X. Mao;A. Rodkina
中科院分区：
文献类型：
--
作者：
J. Appleby;X. Mao;A. Rodkina
考虑不稳定的标量确定性差分方程xn +1}= xn（1+ anf（xn）），n1，x 0 =a.我们展示了如何通过添加随机噪声项$\sigma_ng（x_n）\xi_{n+1}$来稳定这个方程，其中$\xi_n$以概率$1/2$取值+1或-1。我们还证明了一个关于系数有界的标量非线性随机差分方程解的几乎必然渐近稳定性的定理，并展示了随机微分方程的噪声稳定性与该方程的离散化之间的联系。
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 stabilization of a stochastic differential equation, and a discretization of this equation.