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

Abstract

We consider the level hitting times τ_y = 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 τ_y 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 τ_y and M_t: for example, if m(y) = Eτ_y 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.