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We start by providing an explicit characterization and analytical properties,including the persistence phenomena, of the distribution of the extinction time$\mathbb{T}$ of a class of non-Markovian self-similar stochastic processes withtwo-sided jumps that we introduce as a stochastic time-change of Markovianself-similar processes. For a suitably chosen time-changed, we observe, forclasses with two-sided jumps, the following surprising facts. On the one hand,all the $\mathbb{T}$'s within a class have the same law which we identify in asimple form for all classes and reduces, in the spectrally positive case, tothe Fr\'echet distribution. On the other hand, each of its distributioncorresponds to the law of an extinction time of a single Markov process withoutpositive jumps, leaving the interpretation that the time-changed hasannihilated the effect of positive jumps. The example of the non-Markovianprocesses associated to L\'evy stable processes is detailed.

Extinction time of non-Markovian self-similar processes, persistence, annihilation of jumps and the Fr\'echet distribution