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June 2011 Hitting times and the running maximum of Markovian growth-collapse processes
Andreas Löpker, Wolfgang Stadje
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J. Appl. Probab. 48(2): 295-312 (June 2011). DOI: 10.1239/jap/1308662628

Abstract

We consider the level hitting times τy = inf{t ≥ 0 | Xt = y} and the running maximum process Mt = sup{Xs | 0 ≤ st} of a growth-collapse process (Xt)t≥0, defined as a [0, ∞)-valued Markov process that grows linearly between random `collapse' times at which downward jumps with state-dependent distributions occur. We show how the moments and the Laplace transform of τy can be determined in terms of the extended generator of Xt and give a power series expansion of the reciprocal of Ee-sτy. We prove asymptotic results for τy and Mt: for example, if m(y) = Eτy is of rapid variation then Mt / m-1(t) →w 1 as t → ∞, where m-1 is the inverse function of m, while if m(y) is of regular variation with index a ∈ (0, ∞) and Xt is ergodic, then Mt / m-1(t) converges weakly to a Fréchet distribution with exponent a. In several special cases we provide explicit formulae.

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Andreas Löpker. Wolfgang Stadje. "Hitting times and the running maximum of Markovian growth-collapse processes." J. Appl. Probab. 48 (2) 295 - 312, June 2011. https://doi.org/10.1239/jap/1308662628

Information

Published: June 2011
First available in Project Euclid: 21 June 2011

zbMATH: 1229.60104
MathSciNet: MR2840300
Digital Object Identifier: 10.1239/jap/1308662628

Subjects:
Primary: 60K30
Secondary: 60F05, 60J25, 60J75

Rights: Copyright © 2011 Applied Probability Trust

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Vol.48 • No. 2 • June 2011
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