Abstract
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.
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References
Appleby, J.A.D., Buckwar, E.: A constructive comparison technique for determining the asymptotic behaviour of linear functional differential equations with unbounded delay. Differ. Equat. Dyn. Syst. 18 (3), 271–301 (2010)
Appleby, J.A.D., Cheng, J.: On the asymptotic stability of a class of perturbed ordinary differential equations with weak asymptotic mean reversion. Electron. J. Qual. Theory Differ. Equat. Proc. 9th Coll. 1, 1–36 (2011)
Appleby, J.A.D., Mackey, D.: Almost sure polynomial asymptotic stability of scalar stochastic differential equations with damped stochastic perturbations. Electron. J. Qual. Theory Differ. Equat. Proc. 7th Coll. QTDE 2, 1–33 (2004)
Appleby, J.A.D., Patterson, D.D.: Classification of convergence rates of solutions of perturbed ordinary differential equations with regularly varying nonlinearity, 32 pp (2013). arXiv:1303.3345
Appleby, J.A.D., Győri, I., Reynolds, D.W.: On exact rates of decay of solutions of linear systems of Volterra equations with delay. J. Math. Anal. Appl. 320, 56–77 (2006)
Appleby, J.A.D., Rodkina, A., Schurz, H.: Pathwise non-exponential decay rates of solutions of scalar nonlinear stochastic differential equation. Disc. Contin. Dynam. Syst. Ser. B. 6(4), 667–696 (2006)
Appleby, J.A.D., Győri, I., Reynolds, D.W.: On exact convergence rates for solutions of linear systems of Volterra difference equations. J. Differ. Equat. Appl. 12(12), 1257–1275 (2006); Corrigendum: 13(1), 95 (2007)
Appleby, J.A.D., Mackey, D., Rodkina, A.: Almost sure polynomial asymptotic stability for stochastic difference equations. J. Math. Sci. (NY) 149(6), 1629–1647 (2008)
Appleby, J.A.D., Gleeson, J.P., Rodkina, A.: Asymptotic constancy and stability in nonautonomous stochastic differential equations. Cubo 10, 145–159 (2008)
Appleby, J.A.D., Gleeson, J.G., Rodkina, A.: On asymptotic stability and instability with respect to a fading stochastic perturbation. Appl. Anal. 88(4), 579–603 (2009)
Appleby, J.A.D., Mao, X., Wu, H.: On the almost sure partial maxima of solutions of affine stochastic functional differential equations. SIAM J. Math. Anal. 42(2), 646–678 (2010)
Appleby, J.A.D., Cheng, J., Rodkina, A.: Characterisation of the asymptotic behaviour of scalar linear differential equations with respect to a fading stochastic perturbation. Discrete Contin. Dyn. Syst. Suppl. 2011, 79–90 (2011)
Appleby, J.A.D., Cheng, J., Rodkina, A.: On the classification of the asymptotic behaviour of solutions of globally stable scalar differential equations with respect to state–independent stochastic perturbations, 29 pp (2013). arXiv:1310.2343
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation. Encyclopedia of Mathematics and Its Applications, vol. 27. Cambridge University Press, Cambridge (1989)
Evtukhov, V., Samoilenko, A.: Asymptotic representations of solutions of nonautonomous ordinary differential equations with regularly varying nonlinearities. Differ. Equat. 47(5), 627–649 (2011)
Itô, K., McKean, H.P.: Diffusion Processes and Their Sample Paths, 2nd edn. Springer, Berlin (1974)
Kozḿa, A.A.: Asymptotic behavior of one class of solutions of nonlinear nonautonomous second-order differential equations. Nonlinear Oscill. 14(4), 497–511 (2012)
Liu, K.: Some remarks on exponential stability of stochastic differential equations. Stoch. Anal. Appl. 19(1), 59–65 (2001)
Liu, K., Mao, X.: Exponential stability of non-linear stochastic evolution equations. Stoch. Process. Appl. 78, 173–193 (1998)
Liu, K., Mao, X.: Large time behaviour of dynamical equations with random perturbation features. Stoch. Anal. Appl. 19(2), 295–327 (2001)
Mao, X.: Almost sure polynomial stability for a class of stochastic differential equation. Quart. J. Math. Oxford Ser. 43(2), 339–348 (1992)
Mao, X.: Polynomial stability for perturbed stochastic differential equations with respect to semimartingales. Stoch. Process. Appl. 41, 101–116 (1992)
Marić, V.: Regular Variation and Differential Equations. Lecture Notes in Mathematics, vol. 1726. Springer, Berlin (2000)
Shiryaev, A.N: Probability, p. 390, 2nd edn. Springer, New York (1996)
Strauss, A., Yorke, J.A.: On asymptotically autonomous differential equations. Math. Syst. Theory 1, 175–182 (1967)
Strauss, A., Yorke, J.A.: Perturbation theorems for ordinary differential equations. J. Differ. Equat. 3, 15–30 (1967)
Zhang, B., Tsoi, A.H.: Lyapunov functions in weak exponential stability and controlled stochastic systems. J. Ramanujan Math. Soc. 11(2), 85–102 (1996)
Zhang, B., Tsoi, A.H.: Weak exponential asymptotic stability of stochastic differential equations. Stoch. Anal. Appl. 15(4), 643–649 (1997)
Acknowledgements
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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Appleby, J.A.D., Patterson, D.D. (2014). On Necessary and Sufficient Conditions for Preserving Convergence Rates to Equilibrium in Deterministically and Stochastically Perturbed Differential Equations with Regularly Varying Nonlinearity. In: Hartung, F., Pituk, M. (eds) Recent Advances in Delay Differential and Difference Equations. Springer Proceedings in Mathematics & Statistics, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-08251-6_1
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