On Necessary and Sufficient Conditions for Preserving Convergence Rates   to Equilibrium in Deterministically and Stochastically Perturbed Differential   Equations with Regularly Varying Nonlinearity
  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. 
John A.D. Appleby and Denis D. Patterson
math.CA