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

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- an...

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Bibliographic Details
Published inAdvances in applied probability Vol. 51; no. 2; pp. 408 - 424
Main Authors Avram, Florin, Vidmar, Matija
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.06.2019
Applied Probability Trust
Subjects
Binomial distribution
Combinatorics
Continuity (mathematics)
Dividends
Economic models
Markov analysis
Mathematics
Original Article
Probability
Random variables
Random walk
Random walk theory
random walk
scale function
compound binomial risk model
martingale
Primary 60G50
Skip-free Markovian jump process
Secondary 91B30
capital injections 2010 Mathematics Subject Classification: Primary: 60G50
scale functions
random walks
com- pound binomial risk model
dividends
martingales
Secondary: 91B30
skip-free Markovian jump processes
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Summary: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.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 14
ISSN:0001-8678
1475-6064
DOI:10.1017/apr.2019.17