# American Institute of Mathematical Sciences

## Behavior of solution of stochastic difference equation with continuous time under additive fading noise

 Department of Mathematics, Ariel University, Ariel 40700, Israel

Received  July 2020 Revised  November 2020 Published  February 2021

Effect of additive fading noise on a behavior of the solution of a stochastic difference equation with continuous time is investigated. It is shown that if the zero solution of the initial stochastic difference equation is asymptotically mean square quasistable and the level of additive stochastic perturbations is given by square summable sequence, then the solution of a perturbed difference equation remains to be an asymptotically mean square quasitrivial. The obtained results are formulated in terms of Lyapunov functionals and linear matrix inequalities (LMIs). It is noted that the study of the situation, when an additive stochastic noise fades on the infinity not so quickly, remains an open problem.

Citation: Leonid Shaikhet. Behavior of solution of stochastic difference equation with continuous time under additive fading noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021043
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50 trajectories (green) of the solution of the equation (1.1) by $k = 1$, $\tau = 1$, $a_0 = a_1 = b_0 = 0.45$, $b_1 = -0.45$ and $\sigma(t)$ (red)
The same as Fig. 1 besides of $b_1 = 0.45$
The same as Fig. 1 besides of $\sigma(t) = \dfrac{m}{\sqrt{t+\tau-m}}$, $t>m$
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