Hitting times and the running maximum of Markovian growth-collapse processes

A.H. Lopker, W. Stadje

Research output: Contribution to journalArticleAcademicpeer-review

4 Citations (Scopus)

Abstract

We consider the level hitting times ty = inf{t = 0 | Xt = y} and the running maximum process Mt = sup{Xs | 0 = s = t} of a growth-collapse process (Xt)t=0, defined as a [0, 8)-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 ty can be determined in terms of the extended generator of Xt and give a power series expansion of the reciprocal of Ee-sty. We prove asymptotic results for ty and Mt: for example, if m(y) = Ety is of rapid variation then Mt / m-1(t) ¿w 1 as t ¿ 8, where m-1 is the inverse function of m, while if m(y) is of regular variation with index a ¿ (0, 8) 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. Keywords: Growth-collapse process; piecewise deterministic Markov process; hitting time; running maximum; asymptotic behavior; regular variation; separable jump measure
Original languageEnglish
Pages (from-to)295-312
JournalJournal of Applied Probability
Volume48
Issue number2
DOIs
Publication statusPublished - 2011

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