On Necessary and Sufficient Conditions for Preserving Convergence Rates to Equilibrium in Deterministically and Stochastically Perturbed Differential Equations with Regularly Varying Nonlinearity
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This paper develops necessary and sufficient conditions for the preservation of asymptotic convergence rates of deterministically and stochastically perturbed ordinary differential equations with regularly varying nonlinearity close to their equilibrium. Sharp conditions are also established which preserve the asymptotic behaviour of the derivative of the underlying unperturbed equation. Finally, necessary and sufficient conditions are established which enable finite difference approximations to the derivative in the stochastic equation to preserve the asymptotic behaviour of the derivative of the unperturbed equation, even though the solution of the stochastic equation is nowhere differentiable, almost surely.
KeywordsDifferential equations Stochastic differential equations Asymptotic stability Global asymptotic stability State-independent diffusion Fading perturbation Regular variation
John Appleby gratefully acknowledges Science Foundation Ireland for the support of this research under the Mathematics Initiative 2007 grant 07/MI/008 “Edgeworth Centre for Financial Mathematics”. Denis Patterson is supported by the Government of Ireland Postgraduate Scholarship Scheme operated by the Irish Research Council under the project “Persistent and strong dependence in growth rates of solutions of stochastic and deterministic functional differential equations with applications to finance”, GOIPG/2013/402. Both authors thank the referee for their careful review of their manuscript; the first author thanks the organisers of the conference for the opportunity to present his research and the invitation to submit a paper to these proceedings.
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