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On perturbations of an ODE with non-Lipschitz coefficients by a small self-similar noise

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  • Pilipenko, Andrey
  • Proske, Frank Norbert

Abstract

We study the limit behavior of differential equations with non-Lipschitz coefficients that are perturbed by a small self-similar noise. It is proved that the limiting process is equal to the maximal solution or minimal solution with certain probabilities p+ and p−=1−p+, respectively. We propose a space–time transformation that reduces the investigation of the original problem to the study of the exact growth rate of a solution to a certain SDE with self-similar noise. This problem is interesting in itself. Moreover, the probabilities p+ and p− coincide with probabilities that the solution of the transformed equation converges to +∞ or −∞ as t→∞, respectively.

Suggested Citation

  • Pilipenko, Andrey & Proske, Frank Norbert, 2018. "On perturbations of an ODE with non-Lipschitz coefficients by a small self-similar noise," Statistics & Probability Letters, Elsevier, vol. 132(C), pages 62-73.
    • Handle: RePEc:eee:stapro:v:132:y:2018:i:c:p:62-73
    DOI: 10.1016/j.spl.2017.09.005
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    References listed on IDEAS

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    1. Nualart, David & Ouknine, Youssef, 2002. "Regularization of differential equations by fractional noise," Stochastic Processes and their Applications, Elsevier, vol. 102(1), pages 103-116, November.
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