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Discretization of Fractional Fully Nonlinear Equations by Powers of Discrete Laplacians

Author

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  • Indranil Chowdhury

    (Indian Institute of Technology)

  • Espen R. Jakobsen

    (Norwegian University of Science and Technology)

  • Robin Ø Lien

    (Norwegian University of Science and Technology)

Abstract

We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $$\sigma \in (0,2)$$ σ ∈ ( 0 , 2 ) since they involve fractional Laplace operators $$(-\Delta )^{\sigma /2}$$ ( - Δ ) σ / 2 . They arise e.g. in control and game theory as dynamic programming equations – HJB and Isaacs equation – and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $$\sigma $$ σ . The accuracy of previous approximations of fractional fully nonlinear equations depend on $$\sigma $$ σ and are worse when $$\sigma $$ σ is close to 2. We show that the schemes are monotone, consistent, $$L^\infty $$ L ∞ -stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We also prove a second order error bound for smooth solutions and present many numerical examples.

Suggested Citation

  • Indranil Chowdhury & Espen R. Jakobsen & Robin Ø Lien, 2025. "Discretization of Fractional Fully Nonlinear Equations by Powers of Discrete Laplacians," Dynamic Games and Applications, Springer, vol. 15(2), pages 383-405, May.
    • Handle: RePEc:spr:dyngam:v:15:y:2025:i:2:d:10.1007_s13235-024-00601-7
    DOI: 10.1007/s13235-024-00601-7
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    References listed on IDEAS

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    1. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    2. Rama Cont & Ekaterina Voltchkova, 2005. "A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models," Post-Print halshs-00445645, HAL.
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