  In his 1972 paper, John Lamperti characterized all positive self-similar
Markov processes as time-changes of exponentials of Levy processes. In the past
decade the problem of classifying all non-negative self-similar Markov
processes that do not necessarily have zero as a trap has been solved gradually
via connections to ladder height processes and excursion theory. Motivated by a
recent article of Chaumont, Rivero, Panti we classify via jump-type SDEs the
symmetric real-valued self-similar Markov processes that only decrease the
absolute value by jumps and leave zero continuously. Our construction of these
self-similar processes involves a pseudo excursion construction and singular
stochastic calculus arguments ensuring that solutions to the SDEs spend zero
time at zero to avoid problems caused by a "bang-bang" drift.
