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On scale functions for Lévy processes with negative phase-type jumps

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  • Jevgenijs Ivanovs

    (Aarhus University)

Abstract

We provide a novel expression of the scale function for a Lévy process with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical examples suggest that this algorithm allows us to employ phase-type distributions with hundreds of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as matrix-exponential and infinite-dimensional phase-type, can be anticipated.

Suggested Citation

  • Jevgenijs Ivanovs, 2021. "On scale functions for Lévy processes with negative phase-type jumps," Queueing Systems: Theory and Applications, Springer, vol. 98(1), pages 3-19, June.
    • Handle: RePEc:spr:queues:v:98:y:2021:i:1:d:10.1007_s11134-021-09696-w
    DOI: 10.1007/s11134-021-09696-w
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    References listed on IDEAS

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    1. 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.
    2. D'Auria, Bernardo & Kella, Offer & Ivanovs, Jevgenijs & Mandjes, Michel, 2010. "First passage of a Markov additive process and generalized Jordan chains," DES - Working Papers. Statistics and Econometrics. WS ws103923, Universidad Carlos III de Madrid. Departamento de Estadística.
    3. Asmussen, Søren & Avram, Florin & Pistorius, Martijn R., 2004. "Russian and American put options under exponential phase-type Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 79-111, January.
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    Cited by:

    1. Zhang, Aili & Li, Shuanming & Wang, Wenyuan, 2023. "A scale function based approach for solving integral-differential equations in insurance risk models," Applied Mathematics and Computation, Elsevier, vol. 450(C).
    2. Zbigniew Palmowski & Pawe{l} Stk{e}pniak, 2022. "Last passage American cancellable option in L\'evy models," Papers 2212.01119, arXiv.org.

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