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On Maxima and Ladder Processes for a Dense Class of Lévy Process

Part of: Markov processes Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Martijn Pistorius
Martijn Pistorius*
Affiliation:
King's College London
*
∗ Postal address: Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK. Email address: pistoriu@mth.kcl.ac.uk
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Abstract

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In this paper, we present an iterative procedure to calculate explicitly the Laplace transform of the distribution of the maximum for a Lévy process with positive jumps of phase type. We derive error estimates showing that this iteration converges geometrically fast. Subsequently, we determine the Laplace transform of the law of the upcrossing ladder process and give an explicit pathwise construction of this process.

Keywords

Lévy processfirst passageladder processWiener-Hopf factorizationphase-type distributionMarkov additive processnonlinear iteration

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60K15: Markov renewal processes, semi-Markov processes 60J25: Continuous-time Markov processes on general state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 1 , March 2006 , pp. 208 - 220
Copyright
© Applied Probability Trust 2006 

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