Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic Difference Equation

Details Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic Difference Equation Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic Difference Equation

2005-08-19

arxiv.org/abs/math/0508371v3

J. Difference Equ. Appl. 12 (2006) 535-553

Mathematics

Probability

22 pages; corrected more typos, fixed LaTeX macros

Scientific paper

10.1080/10236190600574093

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.

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