On some tractable growth-collapse processes with renewal collapse epochs

Abstract

In this paper we generalize existing results for the steady state distribution of growth collapse processes with independent exponential inter-collapse times to the case where they have a general distribution on the positive real line having a finite mean. In order to compute the moments of the stationary distribution, no further assumptions are needed. However, in order to compute the stationary distribution, the price that we are required to pay is the restriction of the collapse ratio distribution from a general one concentrated on the unit interval to minus-log-phase-type distributions. A random variable has such a distribution if the negative of its natural logarithm has a phase type distribution. Thus, this family of distributions is dense in the family of all distributions concentrated on the unit interval. The approach is to first study a certain Markov modulated shot-noise process from which the steady state distribution for the related growth collapse model can be inferred via level crossing arguments.

Fractional Moments of Solutions to Stochastic Recurrence Equations

“…In this case, B t = A t C t , and A t and B t are dependent for every t with the exception of constant A. The marginal distribution of the solution to (3.4) is known in some particular cases when A and C have gamma-or beta-like distributions; see [13], [14], [15], [16], and [7], [10].…”

“…[17], [23], and [28]. Boxma et al [7] considered (1.1) in the context of growth-collapse processes with renewal collapse epochs. The stochastic recurrence equation (1.1) has also been used in the context of insurance risk models; see [11] and [20].…”

“…In some special cases the distribution of X can be calculated explicitly; see, e.g. [1], [7], [13], [14], [15], [16], [21], [30], and the references therein. Then, in principle, one could also calculate all moments of X, both the integer and the fractional moments.…”

International Reserve Management: A Drift‐switching Reflected Jump‐diffusion Model

“…Moreover, as it is reasonable to believe that jump sizes are typically state dependent, Markovian growth‐collapse models may also serve as suitable alternatives; see, for example, Boxma et al. () and Boxma, Kella, and Perry (). We leave these intriguing issues for future research.…”

Shot-noise queueing models

“…In fact, Kella [61] explicitly pointed out the relation to shot-noise models, stating the following: if X n does not have an atom at zero, and − E logX n < ∞, then the abovedescribed growth-collapse process and the G/G/1 shot-noise queue with interarrival intervals ξ n = −r −1 logX n have the same dynamics just before, and just after, jump epochs. The relation between growth-collapse processes and shot-noise queues is further explored in [24,28]. In the latter paper the intervals between collapses have a general distribution, and the random reduction factor at collapses has a minus-log phase-type distribution, i.e., minus its natural logarithm has a phase-type distribution.…”