Journal
STOCHASTICS-AN INTERNATIONAL JOURNAL OF PROBABILITY AND STOCHASTIC PROCESSES
Volume 81, Issue 2, Pages 99-127Publisher
TAYLOR & FRANCIS LTD
DOI: 10.1080/17442500802088541
Keywords
nonlinear stochastic difference equations; almost sure stability; decay rates; martingale convergence theorem
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Funding
- Dublin City University
- University of the West Indies, Mona
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We consider the stochastic difference equation x(n+1) = x(n)(1 + hf(x(n)) + root hg(x(n))xi(n+1)), n = 0, 1, ... , x(0) is an element of R-1, where f and g are nonlinear, bounded functions, {ji} is a sequence of independent random variables, and h>0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution x(n) equivalent to 0. We also show that, for some natural choices of f and g, the rate of decay of x(n) is approximately polynomial: there exists alpha>0 such that x(n) decays faster than n(-alpha+epsilon) but slower than n(-alpha-epsilon), for any epsilon>0. I t turns out that, if g(x) decays faster than f(x) as x -> 0, the polynomial rate of decay can be established precisely: x(n)n(alpha) tends to a constant limit. On the other hand, if g does not decay quickly enough, the approximate decay rate is the best possible result.
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