Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic Difference Equation
Gregory Berkolaiko, Alexandra Rodkina
Abstract
We consider a non-homogeneous nonlinear stochastic difference equation Xn+1 = Xn (1 + f(Xn)ξn+1) + Sn, and its important special case Xn+1 = Xn (1 + ξn+1) + Sn, both with initial value X0, non-random decaying free coefficient Sn and independent random variables ξn. We establish results on convergence of solutions Xn to zero. The necessary conditions we find tie together certain moments of the noise ξn and the rate of decay of Sn. To ascertain sharpness of our conditions we discuss some situations when Xn diverges. We also establish a result concerning the rate of decay of Xn to zero.
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