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Fast Laplace Transform Methods for Free-Boundary Problems of Fractional Diffusion Equations

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Abstract

In this paper we develop a fast Laplace transform method for solving a class of free-boundary fractional diffusion equations arising in the American option pricing. Instead of using the time-stepping methods, we develop the Laplace transform methods for solving the free-boundary fractional diffusion equations. By approximating the free boundary, the Laplace transform is taken on a fixed space region to replace discretizing the temporal variable. The hyperbola contour integral method is exploited to restore the option values. Meanwhile, the coefficient matrix has theoretically proven to be sectorial. Therefore, the highly accurate approximation by the fast Laplace transform method is guaranteed. The numerical results confirm that the proposed method outperforms the full finite difference methods in regard to the accuracy and complexity.

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Authors and Affiliations

  1. School of Economic Mathematics, Southwestern University of Finance and Economics, Wenjiang, Chengdu, 611130, People’s Republic of China

    Zhiqiang Zhou & Jingtang Ma

  2. Department of Mathematics, University of Macau, Taipa, Macau

    Hai-wei Sun

Authors
  1. Zhiqiang Zhou
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  2. Jingtang Ma
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  3. Hai-wei Sun
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Correspondence to Jingtang Ma.

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Additional information

The work was supported by National Natural Science Foundation of China (Grant No. 11671323), Program for New Century Excellent Talents in University (Grant No. NCET-12-0922) and the Fundamental Research Funds for the Central Universities (Grant No. 15CX141110). It was also partially supported by Research Grants MYRG2016-00063-FST from University of Macau and 054/2015/A2 from FDCT of Macao.

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Zhou, Z., Ma, J. & Sun, Hw. Fast Laplace Transform Methods for Free-Boundary Problems of Fractional Diffusion Equations. J Sci Comput 74, 49–69 (2018). https://doi.org/10.1007/s10915-017-0423-x

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  • DOI: https://doi.org/10.1007/s10915-017-0423-x

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