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)

A Generalized Memoryless Property

“…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] and [8] for similar models.…”

“…This leads to the final solutionf (x) = a + + − a − e a − (x−z * (θ)) + c − a − a + − a − e a + (x−z * (θ)). This example is a generalization of Example (A), Section 4.1 in[8]. Suppose that the jump measure ν(x, y) = s θ (y)/s θ (x) is defined such that for some α > 0 s θ (x)λ(x) = ακ(x).…”

On time reversal of piecewise deterministic Markov processes

“…Piecewise deterministic paths can be observed for Markov processes in a variety of applications. We mention risk process ( [1,48,19,20,25]), growth collapse and stress release models ( [9,53,58,12,41]), queueing models ( [14,11]), earthquake models ( [46]), repairable systems ( [38]), storage models ( [15,27,28]) and TCP data transmission ( [39,40,24]). The mathematical framework that is used in this paper was introduced by Davis [22,21].…”