Hitting times and the running maximum of Markovian growth collapse processes

We consider a Markovian growth collapse process on the state space e = [0;8) which evolves as follows. Between random downward jumps the process increases with slope one. Both the jump intensity and the jump sizes depend on the current state of the process. We are interested in the behavior of the first hitting time Ty = inf{t = 0¦Xt = y} as y becomes large and the growth of the maximum process Mt = sup{Xsj0= s = t} as t¿8. We consider the recursive sequence of equations Amn = mn-1, m0= = 1, where A is the extended generator of the MGCP, and show that the solution sequence (which is essentially unique and can be given in integral form) is related to the moments of Ty. The Laplace transform of Ty can be expressed in closed form (in terms of an integral involving a certain kernel) in a similar way. We derive asymptotic results for the running maximum: (i) if m1(y) is of rapid variation, we have Mt/m -1(t) d¿ 1; (ii) if m1(y) is of regular variation with index a ¿ 2 (0,8) and the MGCP is ergodic, then Mt/m-1(t) d¿ Za, where Za, has a Frechet distribution. We present several examples.