Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes

Abstract

We consider the level hitting times τy= inf{t≥ 0 |Xt=y} and the running maximum processMt= sup{Xs| 0 ≤s≤t} 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 τycan be determined in terms of the extended generator ofXtand give a power series expansion of the reciprocal of Ee−sτy. We prove asymptotic results for τyandMt: for example, ifm(y) = Eτyis of rapid variation thenMt/m-1(t) →w1 ast→ ∞, wherem-1is the inverse function ofm, while ifm(y) is of regular variation with indexa∈ (0, ∞) andXtis ergodic, thenMt/m-1(t) converges weakly to a Fréchet distribution with exponenta. In several special cases we provide explicit formulae.