We consider the level hitting times τy = inf{t ≥ 0 | X t = y} and the running maximum process M t = sup{X s | 0 ≤ s ≤ t} of a growth-collapse process (X t )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 X t and give a power series expansion of the reciprocal of Ee−sτ y . We prove asymptotic results for τy and M t : for example, if m(y) = Eτy is of rapid variation then M t / 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 X t is ergodic, then M t / m -1(t) converges weakly to a Fréchet distribution with exponent a. In several special cases we provide explicit formulae.
