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

A Jump-Type SDE Approach to Real-Valued Self-Similar Markov Processes