Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T07:06:57.497Z Has data issue: false hasContentIssue false
  • English
  • Français

Moments of Markovian growth–collapse processes

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  14 June 2022

Nicolas Privault Open the ORCID record for Nicolas Privault [Opens in a new window]
Nicolas Privault*
Affiliation:
Nanyang Technological University
*
*Postal address: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We apply general moment identities for Poisson stochastic integrals with random integrands to the computation of the moments of Markovian growth–collapse processes. This extends existing formulas for mean and variance available in the literature to closed-form moment expressions of all orders. In comparison with other methods based on differential equations, our approach yields explicit summations in terms of the time parameter. We also treat the case of the associated embedded chain, and provide recursive codes in Maple and Mathematica for the computation of moments and cumulants of any order with arbitrary cut-off moment sequences and jump size functions.

Keywords

Growth–collapse processesPoisson shot noiseuniform cut-off ratesstochastic integrals with jumpsmomentscumulants

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J22: Computational methods in Markov chains 60G55: Point processes
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 4 , December 2022 , pp. 1070 - 1093
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. and Stegun, I. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Vol. 55, 9th edn. Dover, New York.Google Scholar
Boxma, O., Kella, O. and Perry, D. (2011). On some tractable growth–collapse processes with renewal collapse epochs. J. Appl. Prob. 48A, 217–234.CrossRefGoogle Scholar
Boxma, O., Perry, D., Stadje, W. and Zacks, S. (2006). A Markovian growth–collapse model. J. Appl. Prob. 38, 221–243.CrossRefGoogle Scholar
Brémaud, P. (1999). Markov Chains. Springer, New York.CrossRefGoogle Scholar
Chiu, S., Stoyan, D., Kendall, W. and Mecke, J. (2013). Stochastic Geometry and Its Applications, 3rd edn. John Wiley, New York.CrossRefGoogle Scholar
Davis, M. (1984). Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models. J. R. Statist. Soc. B [Statist. Methodology] 46, 353–388.Google Scholar
Daw, A. and Pender, J. (2020). Matrix calculations for moments of Markov processes. Preprint. Available at https://arxiv.org/abs/1909.03320.Google Scholar
Eliazar, I. and Klafter, J. (2004). A growth–collapse model: Lévy inflow, geometric crashes, and generalized Ornstein–Uhlenbeck dynamics. Physica A 334, 1–21.CrossRefGoogle Scholar
Frolkova, M. and Mandjes, M. (2019). A Bitcoin-inspired infinite-server model with a random fluid limit. Stoch. Models 35, 1–32.CrossRefGoogle Scholar
Kella, O. (2009). On growth–collapse processes with stationary structure and their shot-noise counterparts. J. Appl. Prob. 46, 363–371.CrossRefGoogle Scholar
Leonov, V. and Shiryaev, A. (1959). On a method of calculation of semi-invariants. Theory Prob. Appl. 4, 319–329.CrossRefGoogle Scholar
Lukacs, E. (1955). Applications of Faà di Bruno’s formula in mathematical statistics. Amer. Math. Monthly 62, 340–348.Google Scholar
Mecke, J. (1967). Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 36–58.CrossRefGoogle Scholar
Privault, N. (2009). Moment identities for Poisson–Skorohod integrals and application to measure invariance. C. R. Acad. Sci. Paris 347, 1071–1074.CrossRefGoogle Scholar
Privault, N. (2012). Invariance of Poisson measures under random transformations. Ann. Inst. H. Poincaré Prob. Statist. 48, 947–972.CrossRefGoogle Scholar
Privault, N. (2012). Moments of Poisson stochastic integrals with random integrands. Prob. Math. Statist. 32, 227–239.Google Scholar
Privault, N. (2016). Combinatorics of Poisson stochastic integrals with random integrands. In Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener–Itô Chaos Expansions and Stochastic Geometry, eds G. Peccati and M. Reitzner, Springer, Berlin, pp. 37–80.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.CrossRefGoogle Scholar
Slivnyak, I. (1962). Some properties of stationary flows of homogeneous random events. Theory Prob. Appl. 7, 336–341.CrossRefGoogle Scholar