Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noise

Abstract

We consider stochastic difference equationwhere functions f and g are nonlinear and bounded, random variables ξ i are independent and h > 0 is a nonrandom parameter. We establish results on asymptotic stability and instability of the trivial solution x n ≡ 0. We also show, that for some natural choices of the nonlinearities f and g, the rate of decay of x n is approximately polynomial: we find α > 0 such that x n decays faster than n −α+ε but slower than n −α−ε for any ε > 0.It also turns out that if g(x) decays faster than f (x) as x → 0, the polynomial rate of decay can be established exactly, x n n α → const. On the other hand, if the coefficient by the noise does not decay fast enough, the approximate decay rate is the best possible result.1991 Mathematics Subject Classification. 39A10, 39A11, 37H10, 34F05, 93E15.

Corrigendum: On the use of a discrete form of the Itô formula in the article ‘Almost sure asymptotic stability analysis of the -Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations’

“…(2) The proof of the discrete Itô formula developed in [1] must be adapted to the test equation of interest in order to accommodate a perturbation with h-dependent density. (3) There is a general implication for the linear stability analysis of numerical methods for stochastic differential equations: the need to consider more than one test equation in is highlighted in [3,4], and this analysis demonstrates that the discrete Itô formula cannot necessarily be applied to different test equations without adapting the proof to the special structure of each.…”

“…Note that in the original proof in [1] it is assumed for brevity that f and g are non-random constants, and the proof therefore examines the non-conditional expectation, with the comment that the conditional version may be treated similarly.…”

“…Note that terms of the sequence {ζ n } n∈N are independent and identically distributed, and hence the left-hand side of the inequality evaluates to a constant, independent of n. In order to generate from (2.3) a necessary and sufficient condition for almost sure asymptotic stability in terms of the system parameters, a discrete form of the Itô formula, first proved in [1], was applied in [2] to expand the left-hand side of (2.3) for small values of h. This required that the sequence {ζ n } n∈N satisfy the following assumption. Assumption 1.…”

“…To understand the motivation for this article, we must review the proof of Theorem 2.1 in outline; details may be found in [1].…”

Almost sure asymptotic stability analysis of the θ-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations

“…(3) Apply a discrete Itô formula developed in [2] to E[ln(R n )] in order to derive almost sure asymptotic stability and instability conditions (for small step sizes) in terms of the system parameters.…”