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Moments and polynomial expansions in discrete matrix-analytic models

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  • Asmussen, Søren
  • Bladt, Mogens

Abstract

Calculation of factorial moments and point probabilities is considered in integer-valued matrix-analytic models at a finite horizon T. Two main settings are considered, maxima of integer-valued downward skipfree Lévy processes and Markovian point process with batch arrivals (BMAPs). For the moments of the finite-time maxima, the procedure is to approximate the time horizon T by an Erlang distributed one and solve the corresponding matrix Wiener–Hopf factorization problem. For the BMAP, a structural matrix-exponential representation of the factorial moments of N(T) is derived. Moments are then used as a computational vehicle to provide a converging Gram–Charlier series for the point probabilities. Topics such as change-of-measure techniques and time inhomogeneity are also discussed.

Suggested Citation

  • Asmussen, Søren & Bladt, Mogens, 2022. "Moments and polynomial expansions in discrete matrix-analytic models," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 1165-1188.
    • Handle: RePEc:eee:spapps:v:150:y:2022:i:c:p:1165-1188
    DOI: 10.1016/j.spa.2021.12.002
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    References listed on IDEAS

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    1. Anders Hald, 2000. "The Early History of the Cumulants and the Gram‐Charlier Series," International Statistical Review, International Statistical Institute, vol. 68(2), pages 137-153, August.
    2. Carr, Peter, 1998. "Randomization and the American Put," The Review of Financial Studies, Society for Financial Studies, vol. 11(3), pages 597-626.
    3. Søren Asmussen & Patrick J. Laub & Hailiang Yang, 2019. "Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits," Risks, MDPI, vol. 7(1), pages 1-22, February.
    4. Asmussen, Soren & Avram, Florin & Usabel, Miguel, 2002. "Erlangian Approximations for Finite-Horizon Ruin Probabilities," ASTIN Bulletin, Cambridge University Press, vol. 32(2), pages 267-281, November.
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