On asymptotic stability and instability with respect to a fading stochastic perturbation

Abstract

We develop necessary and sufficient conditions for the a.s. asymptotic stability of solutions of a scalar, non-linear stochastic equation with state-independent stochastic perturbations that fade in intensity. These conditions are formulated in terms of the intensity function: roughly speaking, we show that as long as the perturbations fade quicker than some identifiable critical rate, the stability of the underlying deterministic equation is unaffected. These results improve on those of Chan and Williams; for example, we remove the monotonicity requirement on the drift coefficient and relax it on the intensity of the stochastic perturbation. We also employ different analytic techniques.

On Necessary and Sufficient Conditions for Preserving Convergence Rates to Equilibrium in Deterministically and Stochastically Perturbed Differential Equations with Regularly Varying Nonlinearity

“…The Lipschitz continuity of f , the property (1.4), (4.5), and the fact that g is continuous on [0, ∞) and g(t) → 0 as t → ∞ means, by a result in [7], that z(t) → 0 as t → ∞. This allows us to conclude that x(t) → 0 as t → ∞, as claimed.…”

On the asymptotic stability of a class of perturbed ordinary differential equations with weak asymptotic mean reversion

“…Suppose that f obeys (2.2), g obeys (2. 3), and that f obeys (2.22) and g and f obey (2.25). Suppose that x is the unique continuous solution of (2.1).…”

“…in the case when g obeys (2.8) but g ∈ L 1 (0, ∞) and when f obeys (2.2) but the restoring force f (x) as x → ∞ is so weak that lim x→∞ f (x) = 0 (2.11) are presented in Appleby, Gleeson and Rodkina [3]. However, when (2.11) is strengthened so that in addition to (2.2), f also obeys There exists φ > 0 such that φ := lim inf |x|→∞ |f (x)|, (2.12) then the condition (2.8) on g suffices to ensure that the solution x of (2.1) obeys (2.6).…”

“…However, when (2.11) is strengthened so that in addition to (2.2), f also obeys There exists φ > 0 such that φ := lim inf |x|→∞ |f (x)|, (2.12) then the condition (2.8) on g suffices to ensure that the solution x of (2.1) obeys (2.6). See also [3]. For this reason, we restrict our focus in this paper to the case when f obeys (2.11).…”

“…The motivation for this work originates from work on the asymptotic behaviour of stochastic differential equations with state independent perturbations, for which the underlying deterministic equation is globally asymptotically stable. In the case when f has relatively strong mean reversion, it is shown in [3], for a sufficiently rapidly decaying noise intensity, that solutions are still asymptotically stable, but that slower convergence leads to unbounded solutions. A complete categorisation of the asymptotic behaviour in the linear case is given in [1].…”