Hitting times and the Running Maximum of Markovian Growth-collapse Processes

We consider the level hitting times τy = 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,∞)-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 ofXt and give a power series expansion of the reciprocal of E e−sτy . We prove asymptotic results for τy and Mt : for example, if m(y) = E τy is of rapid variation then Mt/m −1(t) w −→ 1 as t → ∞, wherem−1 is the inverse function ofm, while ifm(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.