TY - JOUR

T1 - Hitting times and the running maximum of Markovian growth-collapse processes

AU - Lopker, A.H.

AU - Stadje, W.

PY - 2011

Y1 - 2011

N2 - We consider the level hitting times ty = inf{t = 0 | Xt = y} and the running maximum process Mt = sup{Xs | 0 = s = t} of a growth-collapse process (Xt)t=0, defined as a [0, 8)-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 ty can be determined in terms of the extended generator of Xt and give a power series expansion of the reciprocal of Ee-sty. We prove asymptotic results for ty and Mt: for example, if m(y) = Ety is of rapid variation then Mt / m-1(t) ¿w 1 as t ¿ 8, where m-1 is the inverse function of m, while if m(y) is of regular variation with index a ¿ (0, 8) and Xt is ergodic, then Mt / m-1(t) converges weakly to a Fréchet distribution with exponent a. In several special cases we provide explicit formulae.
Keywords: Growth-collapse process; piecewise deterministic Markov process; hitting time; running maximum; asymptotic behavior; regular variation; separable jump measure

AB - We consider the level hitting times ty = inf{t = 0 | Xt = y} and the running maximum process Mt = sup{Xs | 0 = s = t} of a growth-collapse process (Xt)t=0, defined as a [0, 8)-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 ty can be determined in terms of the extended generator of Xt and give a power series expansion of the reciprocal of Ee-sty. We prove asymptotic results for ty and Mt: for example, if m(y) = Ety is of rapid variation then Mt / m-1(t) ¿w 1 as t ¿ 8, where m-1 is the inverse function of m, while if m(y) is of regular variation with index a ¿ (0, 8) and Xt is ergodic, then Mt / m-1(t) converges weakly to a Fréchet distribution with exponent a. In several special cases we provide explicit formulae.
Keywords: Growth-collapse process; piecewise deterministic Markov process; hitting time; running maximum; asymptotic behavior; regular variation; separable jump measure

U2 - 10.1239/jap/1308662628

DO - 10.1239/jap/1308662628

M3 - Article

VL - 48

SP - 295

EP - 312

JO - Journal of Applied Probability

JF - Journal of Applied Probability

SN - 0021-9002

IS - 2

ER -