On Growth-Collapse Processes with Stationary Structure and Their Shot-Noise Counterparts

Kella, Offer

doi:10.1017/s0021900200005519

Cited by 3 publications

(3 citation statements)

References 9 publications

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“…As soon as the type of move is fixed (down or up), to decide where the process goes precisely, we must use the inverse of the corresponding distribution function (24) (with y ≤ x or y > x), conditioned on the type of move. Remark 14.…”

Section: The Embedded Chainmentioning

confidence: 99%

“…Decay-surge models have been extensively studied in the literature; see among others Eliazar and Klafter [12] and Harrison and Resnick [19,20]. Most studies, however, concentrate on non-Markovian models such as shot-noise or Hawkes processes, where superpositions of overlapping decaying populations are considered; see Brémaud and Massoulié [5], Eliazar and Klafter [13,14], Huillet [23], Kella and Stadje [25], Kella [24], and Brockwell et al [6].…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by storage processes for dams, the papers of Brockwell et al [6], Kella [24], Çinlar and Pinsky [8], Asmussen and Kella [2], Boxma et al [4], Boxma et al [3], and Harrison and Resnick [19] are mostly concerned with growth-collapse models when growth is from stochastic additive inputs such as compound Poisson or Lévy processes or renewal processes.…”

Section: Introductionmentioning

confidence: 99%

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On decay–surge population models

2022

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We consider continuous space–time decay–surge population models, which are semi-stochastic processes for which deterministically declining populations, bound to fade away, are reinvigorated at random times by bursts or surges of random sizes. In a particular separable framework (in a sense made precise below) we provide explicit formulae for the scale (or harmonic) function and the speed measure of the process. The behavior of the scale function at infinity allows us to formulate conditions under which such processes either explode or are transient at infinity, or Harris recurrent. A description of the structures of both the discrete-time embedded chain and extreme record chain of such continuous-time processes is supplied.

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Section: The Embedded Chainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On decay–surge population models

2022

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On exact sampling of stochastic perpetuities

Blanchet

Sigman²

2011

Journal of Applied Probability

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A stochastic perpetuity takes the form D ∞ = ∞ n=0 exp(Y 1 + · · · + Y n )B n , where (Y n : n ≥ 0) and (B n : n ≥ 0) are two independent sequences of independent and identically distributed random variables (RVs). This is an expression for the stationary distribution of the Markov chain defined recursively bywhere A and B are independent copies of the A n and B n (and independent of D ∞ on the right-hand side). In our framework, the quantity B n , which represents a random reward at time n, is assumed to be positive, unbounded with EB p n < ∞ for some p > 0, and have a suitably regular continuous positive density. The quantity Y n is assumed to be light tailed and represents a discount rate from time n to n − 1. The RV D ∞ then represents the net present value, in a stochastic economic environment, of an infinite stream of stochastic rewards. We provide an exact simulation algorithm for generating samples of D ∞ . Our method is a variation of dominated coupling from the past and it involves constructing a sequence of dominating processes.

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On some tractable growth-collapse processes with renewal collapse epochs

Boxma

Kella²,

Perry³

2011

Journal of Applied Probability

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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.

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On Growth-Collapse Processes with Stationary Structure and Their Shot-Noise Counterparts

Cited by 3 publications

References 9 publications

On decay–surge population models

On decay–surge population models

On exact sampling of stochastic perpetuities

On some tractable growth-collapse processes with renewal collapse epochs

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