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

摘要

We consider the level hitting times ι_y = inf {t > 0 | X_t = y} and the running maximum process Mt = sup{X_s | 0 < s < t} of a growth-collapse process (X_t)_t>o, 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 ry can be determined in terms of the extended generator of X_t and give a power series expansion of the reciprocal of Ee—~(sιy) .We prove asymptotic results for ι_y and M_t: for example, if m(y) = E ι_y is of rapid variation then M_t / m~(-1) (t)→ 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 Frechet distribution with exponent a. In several special cases we provide explicit formulae.