Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic   Difference Equation
  We consider a non-homogeneous nonlinear stochastic difference equation   X_{n+1} = X_n (1 + f(X_n)\xi_{n+1}) + S_n, and its important special case   X_{n+1} = X_n (1 + \xi_{n+1}) + S_n, both with initial value X_0, non-random decaying free coefficient S_n and independent random variables \xi_n. We establish results on \as convergence of solutions X_n to zero. The necessary conditions we find tie together certain moments of the noise \xi_n and the rate of decay of S_n. To ascertain sharpness of our conditions we discuss some situations when X_n diverges. We also establish a result concerning the rate of decay of X_n to zero. 
Gregory Berkolaiko, Alexandra Rodkina