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Markov chain approximations to scale functions of Lévy processes

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  • Mijatović, Aleksandar
  • Vidmar, Matija
  • Jacka, Saul

Abstract

We introduce a general algorithm for the computation of the scale functions of a spectrally negative Lévy process X, based on a natural weak approximation of X via upwards skip-free continuous-time Markov chains with stationary independent increments. The algorithm consists of evaluating a finite linear recursion with its (nonnegative) coefficients given explicitly in terms of the Lévy triplet of X. Thus it is easy to implement and numerically stable. Our main result establishes sharp rates of convergence of this algorithm providing an explicit link between the semimartingale characteristics of X and its scale functions, not unlike the one-dimensional Itô diffusion setting, where scale functions are expressed in terms of certain integrals of the coefficients of the governing SDE.

Suggested Citation

  • Mijatović, Aleksandar & Vidmar, Matija & Jacka, Saul, 2015. "Markov chain approximations to scale functions of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3932-3957.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:10:p:3932-3957
    DOI: 10.1016/j.spa.2015.05.012
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    References listed on IDEAS

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    1. Baurdoux, E.J. & Palmowski, Z. & Pistorius, M.R., 2017. "On future drawdowns of Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2679-2698.
    2. Baurdoux, Erik J. & Palmowski, Z & Pistorius, Martijn R, 2017. "On future drawdowns of Lévy processes," LSE Research Online Documents on Economics 84342, London School of Economics and Political Science, LSE Library.

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