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Exit problems for positive self-similar Markov processes with one-sided jumps
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A systematic exposition of scale functions is given for positive self-similar Markov processes (pssMp) with one-sided jumps. The scale functions express as convolution series of the usual scale functions associated with spectrally one-sided L\'evy processes that underly the pssMp through the Lamperti transform. This theory is then brought to bear on solving the spatio-temporal: (i) two-sided exit problem; (ii) joint first passage problem upwards for the the pssMp and its multiplicative drawdown (resp. drawup) in the spectrally negative (resp. positive) case.
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