Alexandra Rodkina and Henri Schurz
A Martingale Approach to Asymptotic Stability Of Nonlinear Stochastic Difference Equations With Bounded Noise in R¹
Dynamic Systems and Applications 32 (2023) 129-155

https://doi.org/10.46719/dsa2023.32.08

ABSTRACT.
Necessary and sufficient conditions for almost sure asymptotic stability of solutions of stochastic dynamical systems generated by linear and nonlinear, nonautonomous ordinary stochastic difference equations (SDE) in R¹ .

X_n+1    =   X_n  (1 – α_n f(X_n)  +  σ_n g(X_n ) ξ_n+1 )

driven by square-integrable independent random variables ( ξ_n+1)_{n ϵ N}  with uniformly bounded quantities σ_nξ_n+1 are in the center of this presentation. All conditions are explicitly expressed in terms of the coefficients α_n, σ_n, f and g. Kolmogorov’s variant of the strong law of large numbers as well as martingale convergence and martingale representation theorems are applied to prove related results.

AMS (MOS) Subject Classication. 34F05, 37H10, 60H10, 60H25, 60H35, 65C30, 93E15.