HITTING TIMES AND THE RUNNING MAXIMUM OF MARKOVIAN GROWTH-COLLAPSE PROCESSES

Abstract

We consider the level hitting times tau(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, infinity)-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 tau(y) can be determined in terms of the extended generator of X-y and give a power series expansion of the reciprocal of E e(y.)(-s tau) We prove asymptotic results for tau(y) and M-t: for example, if in (y) = E tau(y) is of rapid variation then M-t/m(-1) (t) (w) under right arrow 1 as t -> infinity, where m(-1) is the inverse function of m, while if m(y) is of regular variation with index a is an element of (0, infinity) and X-t is ergodic, then M-t/m(-1) (t) converges weakly to a Frechet distribution with exponent a. In several special cases we provide explicit formulae.