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Approximation methods for piecewise deterministic Markov processes and their costs

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  • Peter Kritzer
  • Gunther Leobacher
  • Michaela Szolgyenyi
  • Stefan Thonhauser

Abstract

In this paper, we analyse piecewise deterministic Markov processes, as introduced in Davis (1984). Many models in insurance mathematics can be formulated in terms of the general concept of piecewise deterministic Markov processes. In this context, one is interested in computing certain quantities of interest such as the probability of ruin of an insurance company, or the insurance company's value, defined as the expected discounted future dividend payments until the time of ruin. Instead of explicitly solving the integro-(partial) differential equation related to the quantity of interest considered (an approach which can only be used in few special cases), we adapt the problem in a manner that allows us to apply deterministic numerical integration algorithms such as quasi-Monte Carlo rules; this is in contrast to applying random integration algorithms such as Monte Carlo. To this end, we reformulate a general cost functional as a fixed point of a particular integral operator, which allows for iterative approximation of the functional. Furthermore, we introduce a smoothing technique which is applied to the integrands involved, in order to use error bounds for deterministic cubature rules. On the analytical side, we prove a convergence result for our PDMP approximation, which is of independent interest as it justifies phase-type approximations on the process level. We illustrate the smoothing technique for a risk-theoretic example, and provide a comparative study of deterministic and Monte Carlo integration.

Suggested Citation

  • Peter Kritzer & Gunther Leobacher & Michaela Szolgyenyi & Stefan Thonhauser, 2017. "Approximation methods for piecewise deterministic Markov processes and their costs," Papers 1712.09201, arXiv.org, revised Jan 2019.
    • Handle: RePEc:arx:papers:1712.09201
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    References listed on IDEAS

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    1. Andreas Eichler & Gunther Leobacher & Michaela Szolgyenyi, 2016. "Utility Indifference Pricing of Insurance Catastrophe Derivatives," Papers 1607.01110, arXiv.org, revised May 2017.
    2. Katia Colaneri & Zehra Eksi & Rudiger Frey & Michaela Szolgyenyi, 2016. "Optimal Liquidation under Partial Information with Price Impact," Papers 1606.05079, arXiv.org, revised Jun 2019.
    3. Siegl, Thomas & F. Tichy, Robert, 2000. "Ruin theory with risk proportional to the free reserve and securitization," Insurance: Mathematics and Economics, Elsevier, vol. 26(1), pages 59-73, February.
    4. B de Saporta & F Dufour & H Zhang & C Elegbede, 2012. "Optimal stopping for the predictive maintenance of a structure subject to corrosion," Journal of Risk and Reliability, , vol. 226(2), pages 169-181, April.
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    Cited by:

    1. Simon Pojer & Stefan Thonhauser, 2023. "The Markovian Shot-noise Risk Model: A Numerical Method for Gerber-Shiu Functions," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-26, March.
    2. Stefan Kremsner & Alexander Steinicke & Michaela Szölgyenyi, 2020. "A Deep Neural Network Algorithm for Semilinear Elliptic PDEs with Applications in Insurance Mathematics," Risks, MDPI, vol. 8(4), pages 1-18, December.
    3. Josef Anton Strini & Stefan Thonhauser, 2020. "On Computations in Renewal Risk Models—Analytical and Statistical Aspects," Risks, MDPI, vol. 8(1), pages 1-20, March.
    4. Stefan Kremsner & Alexander Steinicke & Michaela Szolgyenyi, 2020. "A deep neural network algorithm for semilinear elliptic PDEs with applications in insurance mathematics," Papers 2010.15757, arXiv.org, revised Dec 2020.

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