Exit problems for positive self-similar Markov processes with one-sided jumps

TL;DR

This paper develops a comprehensive theory of scale functions for positive self-similar Markov processes with one-sided jumps, enabling solutions to key exit problems and joint passage problems through convolution series representations.

Contribution

It introduces a systematic exposition of scale functions expressed as convolution series, facilitating analysis of exit and passage problems for pssMp with one-sided jumps.

Findings

Derived convolution series expressions for scale functions.

Solved two-sided exit problems for pssMp.

Addressed joint first passage problems in spectrally negative and positive cases.

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 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.