Discretization of Fractional Fully Nonlinear Equations by Powers of Discrete Laplacians

Indranil Chowdhury (), Espen R. Jakobsen () and Robin Ø Lien ()
Additional contact information
Indranil Chowdhury: Indian Institute of Technology
Espen R. Jakobsen: Norwegian University of Science and Technology
Robin Ø Lien: Norwegian University of Science and Technology

Dynamic Games and Applications, 2025, vol. 15, issue 2, No 3, 383-405

Abstract: 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.

Keywords: Fractional and nonlocal equations; Fully nonlinear equation; HJB equations; Isaacs equations; Degenerate equation; Stochastic control; Lévy processes; Convergence; Error bound; Viscosity solution; Numerical method; Monotone scheme; Powers of discrete Laplacians; 49L25; 35J60; 34K37; 35R11; 35J70; 45K05; 49L25; 49M25; 93E20; 65N06; 65M15; 65R20; 65N12 (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13235-024-00601-7 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:dyngam:v:15:y:2025:i:2:d:10.1007_s13235-024-00601-7

Ordering information: This journal article can be ordered from
http://www.springer.com/economics/journal/13235

DOI: 10.1007/s13235-024-00601-7

Access Statistics for this article

Dynamic Games and Applications is currently edited by Georges Zaccour

More articles in Dynamic Games and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().