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Conformal Accelerations Method and Efficient Evaluation of Stable Distributions

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Abstract

We suggest 3 families of conformal deformations and changes of variables for evaluation of integrals arising in applications of the Fourier analysis to fractional partial differential equations and evaluation of special functions, probability distribution functions, cumulative probability distribution functions and quantiles of stable distributions. For the error tolerance E-15, hypergeometric functions can be calculated much faster (in Matlab implementation) than using SFT in Matlab, Python and Mathematica; even when the index \(\alpha \) of the stable distribution is small or close to 1, the same error tolerance can be satisfied in 0.005–0.1 msec. For the calculation of quantiles in wide regions in the tails using the Newton or bisection method, it suffices to precompute several hundred values of the characteristic exponent at points of an appropriate grid (conformal principal components) and use these values in formulas for cpdf and pdf. The same three families can be used to evaluate more general distributions and solutions of boundary problems for fractional partial differential equations more general than the ones related to stable distributions. The methods of the paper are applicable to other classes of integrals, highly oscillatory ones especially.

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Notes

  1. We are grateful to the anonymous referee for the suggestion to clarify the relation of this definition and notation to the definitions of the Hardy space and norm used in the literature.

  2. Statements of the form “the following choice is approximately optimal” in the paper can be formulated as Lemmas and “the level of optimality” characterized exactly. Typically, “the optimal” choices are impossible. The recommendations which are give are supported by our numerical experiments.

  3. The tails of the distributions decay too slowly, hence, the Monte Carlo simulations are moderately efficient only if the index of the process is close to 2, and the distribution does not differ much from the normal distribution, with the exception of far parts of the tails, which can be safely disregarded in this case.

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Acknowledgements

We acknowledge valuable suggestions for improvements of the paper made by Chief Editors John King and Benoit Perthame, and two anonymous referees. The usual disclaimer applies.

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Authors and Affiliations

  1. Department of Economics, The University of Texas at Austin, 2225 Speedway Stop C3100, Austin, TX, 78712, USA

    Svetlana Boyarchenko

  2. Calico Science Consulting, Austin, TX, USA

    Sergei Levendorskiĭ

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  1. Svetlana Boyarchenko
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  2. Sergei Levendorskiĭ
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Correspondence to Svetlana Boyarchenko.

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Boyarchenko, S., Levendorskiĭ, S. Conformal Accelerations Method and Efficient Evaluation of Stable Distributions. Acta Appl Math 169, 711–765 (2020). https://doi.org/10.1007/s10440-020-00320-2

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