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

Part of: Mathematical economics Stochastic processes

Published online by Cambridge University Press:  07 August 2019

Florin Avram  and
Matija Vidmar
Florin Avram*
Affiliation:
University of Pau
Matija Vidmar*
Affiliation:
University of Ljubljana
*
*Postal address: Laboratoire de Mathématiques Appliquées, Université de Pau, Avenue de l’Université, 64012 Pau, France.
**Postal address: Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 21, 1000 Ljubljana, Slovenia.
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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.

Keywords

Skip-free Markovian jump processrandom walkscale functionmartingalecompound binomial risk model

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 91B30: Risk theory, insurance
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 2 , June 2019 , pp. 408 - 424
Copyright
© Applied Probability Trust 2019 

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