39 research outputs found
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.Comment: 22 pages; corrected more typos, fixed LaTeX macro