Abstract
We revisit the zero-noise Peano selection problem for Levy-driven stochastic differential equation considered in [Pilipenko and Proske, Statist. Probab. Lett., 132:62–73, 2018] and show that the selection phenomenon pertains in the multiplicative noise setting and is robust with respect to certain perturbations of the irregular drift and of the small jumps of the noise.
Highlights
We revisit the zero-noise Peano selection problem for Lévy-driven stochastic differential equation considered in [Pilipenko and Proske, Statist
Lett., 132:62–73, 2018] and show that the selection phenomenon pertains in the multiplicative noise setting and is robust with respect to certain perturbations of the irregular drift and of the small jumps of the noise
Solutions of the small noise stochastic differential equations (SDE) should converge to one of the various deterministic solutions and the selection problem consists in description of this limit behaviour
Summary
We revisit the zero-noise Peano selection problem for Lévy-driven stochastic differential equation considered in [Pilipenko and Proske, Statist. Let us consider an SDE with a drift a and assume that the underlying ODE dx = a(x) dt has multiple solutions. It was shown that the limit law Law(Xε|Xε(0) = 0) is supported by the deterministic maximal and minimal solutions of the ODE dx = a(x) dt starting at zero with the selection probabilities p± that can be explicitly determined, see [1, Theorem 4.1].
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