Exit Problems for Positive Self-Similar Markov Processes with One-Sided Jumps

Abstract

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 Lévy 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 for the pssMp and its multiplicative drawdown (resp. drawup) in the spectrally negative (resp. positive) case.