On scale functions for Lévy processes with negative phase-type jumps

Jevgenijs Ivanovs

doi:10.1007/s11134-021-09696-w

On scale functions for Lévy processes with negative phase-type jumps

^*Corresponding author for this work

Department of Mathematics

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7 Citations (Scopus)

Abstract

We provide a novel expression of the scale function for a Lévy process with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical examples suggest that this algorithm allows us to employ phase-type distributions with hundreds of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as matrix-exponential and infinite-dimensional phase-type, can be anticipated.

Original language	English
Journal	Queueing Systems
Volume	98
Issue	1-2
Pages (from-to)	3-19
Number of pages	17
ISSN	0257-0130
DOIs	https://doi.org/10.1007/s11134-021-09696-w
Publication status	Published - Mar 2021

Keywords

Fluid flow model
Iterative scheme
Phase-type distribution
Rational transform
Scale function

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10.1007/s11134-021-09696-w

https://arxiv.org/pdf/2012.08380.pdf

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abstract = "We provide a novel expression of the scale function for a L{\'e}vy process with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical examples suggest that this algorithm allows us to employ phase-type distributions with hundreds of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as matrix-exponential and infinite-dimensional phase-type, can be anticipated.",

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N2 - We provide a novel expression of the scale function for a Lévy process with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical examples suggest that this algorithm allows us to employ phase-type distributions with hundreds of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as matrix-exponential and infinite-dimensional phase-type, can be anticipated.

AB - We provide a novel expression of the scale function for a Lévy process with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical examples suggest that this algorithm allows us to employ phase-type distributions with hundreds of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as matrix-exponential and infinite-dimensional phase-type, can be anticipated.

KW - Fluid flow model

KW - Iterative scheme

KW - Phase-type distribution

KW - Rational transform

KW - Scale function

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ER -

On scale functions for Lévy processes with negative phase-type jumps

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Keywords

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