Hitting Times and the Running Maximum of Markovian Growth-Collapse Processes

Abstract

We consider a Markovian growth collapse process on the state space E = [0, ∞) 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 τ y = inf{t ≥ 0|X t = y} as y becomes large and the growth of the maximum process M t = sup{X s |0 ≤ s ≤ t} as t → ∞. We consider the recursive sequence of equations Am n = m n−1 , m 0 ≡ 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 τ y . The Laplace transform of τ y 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 m 1 (y) is of rapid variation, we have M t /m −1 (t)

Unified approach for solving exit problems for additive-increase and multiplicative-decrease processes

“…Note that this result matches the result reported in [37,Section 4.2], in which the authors instead use a martingale approach for the derivation of the recursive equation satisfied by the Laplace-Stieltjes transform of the exit time.…”

“…This proves (36) for a > c/p. For a ≤ c/p, instead, we note that E pa [e −wτ ā ↓c (pa) ] = 1, so that now (37) becomes…”

A Generalized Memoryless Property

“…The following lemma generalizes formula (28) in [8]. Proof: The process X t , if started in the state x ∈ I θ will leave I θ only at the moment when it passes through the upper boundary z * (θ ) and ν(x, y) = 0 for y < z * (θ ).…”

“…In this section we consider a piecewise deterministic Markov process X t with jumps that are governed by a jump measure with the generalized lack of memory property described above. See [2,8] for similar models.…”

Piecewise linear process with renewal starting points

“…Remark 3.2. The mean of the first passage time T x (y) for a Markovian growth-collapse process has been studied in detail, see Löpker and Stadje (2011). Meanwhile, formula (3.3) presents the explicit representation of the distribution (in the special case of two-state processes with exponentially distributed jumps).…”