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First passage problems for upwards skip-free random walks via the scale functions paradigm

Published online by Cambridge University Press:  07 August 2019

Florin Avram
Affiliation:
University of Pau
Matija Vidmar
Affiliation:
University of Ljubljana

Abstract

In this paper we develop the theory of the W and Z scale functions for right-continuous (upwards skip-free) discrete-time, discrete-space random walks, along the lines of the analogous theory for spectrally negative Lévy processes. Notably, we introduce for the first time in this context the one- and two-parameter scale functions Z, which appear for example in the joint deficit at ruin and time of ruin problems of actuarial science. Comparisons are made between the various theories of scale functions as one makes time and/or space continuous.

Type
Original Article
Copyright
© Applied Probability Trust 2019 

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