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We study a class of self-similar jump type SDEs driven by H\"older-continuousdrift and noise coefficients. Using the Lamperti transformation for positiveself-similar Markov processes we obtain a necessary and sufficient conditionfor almost sure extinction in finite time. We then show that for certainparameters pathwise uniqueness holds in a restricted sense, namely amongsolutions spending a Lebesgue-negligible amount of time at 0. A direct powertransformation plays a key role.

Hitting properties and non-uniqueness for SDE driven by stable processes