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Nonlinear Parabolic Equations Arising in Mathematical Finance

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Novel Methods in Computational Finance

Part of the book series: Mathematics in Industry ((TECMI,volume 25))

Abstract

This survey chapter is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models of stochastic dynamic portfolio optimization leading to the Hamilton-Jacobi-Bellman (HJB) equation. After suitable transformations, both problems can be represented by solutions to nonlinear parabolic equations. Qualitative analysis will be focused on issues concerning the existence and uniqueness of solutions. In the numerical part we discuss a stable finite-volume and finite difference schemes for solving fully nonlinear parabolic equations.

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Acknowledgements

This research was supported by the European Union in the FP7-PEOPLE-2012-ITN project STRIKE—Novel Methods in Computational Finance (304617).

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Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Comenius University, 842 48, Bratislava, Slovakia

    Daniel Ševčovič

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  1. Daniel Ševčovič
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Correspondence to Daniel Ševčovič .

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Editors and Affiliations

  1. Lehrstuhl für Angewandte Mathematik/Numerische Analysis, Bergische Universität Wuppertal, Wuppertal, Germany

    Matthias Ehrhardt

  2. Lehrstuhl für Angewandte Mathematik/Numerische Analysis, Bergische Universität Wuppertal, Wuppertal, Germany

    Michael Günther

  3. Lehrstuhl für Angewandte Mathematik/Numerische Analysis, Bergische Universität Wuppertal, Wuppertal, Germany

    E. Jan W. ter Maten

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Ševčovič, D. (2017). Nonlinear Parabolic Equations Arising in Mathematical Finance. In: Ehrhardt, M., Günther, M., ter Maten, E. (eds) Novel Methods in Computational Finance. Mathematics in Industry(), vol 25. Springer, Cham. https://doi.org/10.1007/978-3-319-61282-9_1

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