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Multilayer heat equations and their solutions via oscillating integral transforms

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  • Andrey Itkin
  • Alexander Lipton
  • Dmitry Muravey

Abstract

By expanding the Dirac delta function in terms of the eigenfunctions of the corresponding Sturm-Liouville problem, we construct some new (oscillating) integral transforms. These transforms are then used to solve various finance, physics, and mathematics problems, which could be characterized by the existence of a multilayer spatial structure and moving (time-dependent) boundaries (internal interfaces) between the layers. Thus, constructed solutions are semi-analytical and extend the authors' previous work (Itkin, Lipton, Muravey, Multilayer heat equations: application to finance, FMF, 1, 2021). However, our new method doesn't duplicate the previous one but provides alternative representations of the solution which have different properties and serve other purposes.

Suggested Citation

  • Andrey Itkin & Alexander Lipton & Dmitry Muravey, 2021. "Multilayer heat equations and their solutions via oscillating integral transforms," Papers 2112.00949, arXiv.org, revised Dec 2021.
    • Handle: RePEc:arx:papers:2112.00949
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    References listed on IDEAS

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    1. Antoine Lejay & Paolo Pigato, 2019. "A Threshold Model For Local Volatility: Evidence Of Leverage And Mean Reversion Effects On Historical Data," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-24, June.
    2. Paolo Pigato, 2019. "Extreme at-the-money skew in a local volatility model," Finance and Stochastics, Springer, vol. 23(4), pages 827-859, October.
    3. Carr, Elliot J. & March, Nathan G., 2018. "Semi-analytical solution of multilayer diffusion problems with time-varying boundary conditions and general interface conditions," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 286-303.
    4. Marc Decamps & Marc Goovaerts & Wim Schoutens, 2006. "Self Exciting Threshold Interest Rates Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(07), pages 1093-1122.
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