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

Andrey Pilipenko and Frank Norbert Proske

Statistics & Probability Letters, 2018, vol. 132, issue C, 62-73

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.

Keywords: Zero-noise limit; Peano phenomenon; Growth rate of solutions to stochastic equations; Selection of a solution; Self-similar process (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1016/j.spl.2017.09.005

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